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Deflating the Dimensionality Curse in Electronic Structure Calculations

Permanent Link: http://ufdc.ufl.edu/UFE0045227/00001

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Title: Deflating the Dimensionality Curse in Electronic Structure Calculations
Physical Description: 1 online resource (124 p.)
Language: english
Creator: Melnichuk, Anna
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

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Subjects / Keywords: ccsd -- chemistry -- cluster -- coupled -- quantum
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many-body problems whether classical or quantum suffer from what is known as a dimensionality curse. In solving physical problems, we are generally interested in quantifying interactions between physical entities and as the quantity of these entities increases, the number of interaction equations one must solve increases much faster. The work described in this dissertation presents several solutions for the reduction of the dimension of calculations of electronic structure of molecules. Techniques and examples are drawn from calculations of excited states, UV/Vis absorption cross sections, and high-level theoretical treatment of multi-reference electronic structure problems.  The techniques described here may be combined to achieve highly accurate theoretical results for challenging examples at only a small fraction of the cost. Speed-up upwards of 500x were observed for several systems versus the established methods and several examples are presented which would not be otherwise practical to do given the current computational limitations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Anna Melnichuk.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Bartlett, Rodney J.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045227:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045227/00001

Material Information

Title: Deflating the Dimensionality Curse in Electronic Structure Calculations
Physical Description: 1 online resource (124 p.)
Language: english
Creator: Melnichuk, Anna
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: ccsd -- chemistry -- cluster -- coupled -- quantum
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many-body problems whether classical or quantum suffer from what is known as a dimensionality curse. In solving physical problems, we are generally interested in quantifying interactions between physical entities and as the quantity of these entities increases, the number of interaction equations one must solve increases much faster. The work described in this dissertation presents several solutions for the reduction of the dimension of calculations of electronic structure of molecules. Techniques and examples are drawn from calculations of excited states, UV/Vis absorption cross sections, and high-level theoretical treatment of multi-reference electronic structure problems.  The techniques described here may be combined to achieve highly accurate theoretical results for challenging examples at only a small fraction of the cost. Speed-up upwards of 500x were observed for several systems versus the established methods and several examples are presented which would not be otherwise practical to do given the current computational limitations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Anna Melnichuk.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Bartlett, Rodney J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045227:00001


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DEFLATINGTHEDIMENSIONALITYCURSEINELECTRONICSTRUCTUR E CALCULATIONS By ANNMELNICHUK ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c r 2013AnnMelnichuk 2

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Tomyfriendsandlovedones 3

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ACKNOWLEDGMENTS IamdeeplygratefulforthehelpanddirectionofDr.RodneyJ .Bartlettwhohas beenmydissertationadviserforthepastsixyears.Ithanko urgraduatedgroup members:Andrew,TomH.,Josh,andPrakashfromwhomIhavele arnedmuchand Ithankourcurrentgroupmembers:Victor,TomW.,Robert,Al exandreandMattwith whomI'vehadmanystimulatingconversations.Myspecialth ankstoDmitrywhohasnot onlybeenaninvaluableresourcebutagreatfriend. IthankAjithfortirelesslyteachingourgraduatestudents howtobecomebetter scientists,programmers,andhumanbeings.IthankCharlie ,JonandCraigfortheir effortsinrunningtheUFHPCcenterandteachingmemuchabou tservermaintenance andtheLinuxenvironment. ManythanksgoouttomycommitteememberswithwhomI'vehadm anyinteresting discussionsandcollaborationsovertheyears:Dr.Deumens ,Dr.Bowers,Dr.Brucatand Dr.Sullivan.Finally,I'dliketoextendmygratitudetothe QTPmembersaswellastothe membersoftheChemistryandtheElectricalandComputerEng ineeringdepartments fortheirencouragementandsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................7 LISTOFFIGURES .....................................8 ABSTRACT .........................................10 CHAPTER 1INTRODUCTIONTODIMENSIONALITYISSUESINQUANTUMMECHANI CS 11 1.1Hatree-Fockreference .............................11 1.2Dimensionalityissuesinelectronicstructure .................13 1.2.1Groundstate ..............................13 1.2.2Excitedstate ..............................18 1.3Dimensionalityissuesinmolecularstructure .................20 1.4Implementationtechniquesandsoftwareused ...............21 2CHARGETRANSFEREXCITEDSTATES .....................22 2.1Background ...................................22 2.2Backgroundonexamples ...........................22 2.3Methods .....................................25 2.4Resultsanddiscussion ............................26 2.4.1 Abinitio spectrumofphenol ......................26 2.4.2 Abinitio spectraof -naphtholand -naphthol ............34 2.4.3Comparisontoexperimentalresults ..................40 2.5Conclusion ...................................42 3ABSORPTIONCROSSSECTION .........................45 3.1Background ...................................45 3.2Methods .....................................46 3.2.1Approachtotheproblem ........................46 3.2.2Discretevariablerepresentation ....................47 3.2.3Discreteabsorptioncrosssection ...................48 3.2.4CombineDVRandthediscreteabsorptioncrosssection ......49 4ABSORPTIONCROSSSECTIONEXAMPLES ..................52 4.1SodiumHydroxide ...............................52 4.1.1Electronicstructurecalculations ....................52 4.1.2AbsorptionCrossSectionModel ...................55 4.1.2.1Dissociativecoordinateastheprimarycoordinate ....55 5

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4.1.2.2Computetheabsorptioncrosssectionoftheprimary mode .............................57 4.1.2.3Considersignicantlythermallypopulatedvibrat ions ...57 4.1.2.4Determinetheimpactofeachsecondarymode ......59 4.1.2.5Addtheeffectsofotherdegreesoffreedom ........62 4.1.2.6Temperatureeffectscanbeintroducedatthisstage ...64 4.2Water:exampleofaboundsystem ......................65 5SINGLEREFERENCECCFORMULTI-REFERENCEPROBLEMS ......69 5.1Background ...................................69 5.2Tailored-CCextension .............................69 5.3Extendedspacechoice ............................72 5.4Statisticalanalysistechniques .........................75 5.4.1Quantile-quantileplot ..........................75 5.4.2Kerneldensityestimation .......................75 6ACTIVESPACECCEXAMPLES ..........................77 6.1Hydrogenuoride ...............................77 6.1.1SourceofNPEinTCCSD .......................80 6.1.2FXTCCSD ................................84 6.1.3Activespacechoice ..........................84 6.2Fluorinemolecule ...............................88 6.3Ethylene .....................................94 6.3.1Background ...............................94 6.3.2Results .................................96 6.4Bicyclo[1,1,0]butane ..............................99 6.4.1Background ...............................99 6.4.2Results .................................101 6.5FrequenciesandZPE .............................104 6.6Conclusion ...................................106 7CONCLUSIONSANDFUTUREWORK ......................107 REFERENCES .......................................109 BIOGRAPHICALSKETCH ................................124 6

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LISTOFTABLES Table page 2-1Geometryofphenolandphenol-NH 3 ........................28 2-2Verticalabsorptionspectrumofphenolandphenol-NH 3 .............29 2-3Groundstatenaphtholenergy ............................35 2-4Geometryofnaphtholandnaphthol-NH 3 ......................36 2-5Excitationenergiesandpropertiesof and naphthol ..............40 2-6Excitationenergiesofthersttwoexcitedstates .................41 2-7Solvatochromicshiftsofexcitationenergies ....................43 4-1CalculatedexcitedstatesforNaOH .........................52 4-2Near-UVEOM-CCSDexcitationenergiesforNaOH ...............54 4-3Near-UVSTEOMexcitationenergiesforNaOH ..................54 4-4alculatedharmonicandanharmonicvibrationalmodesfo rNaOH ........58 4-5BendingmodeimpactonexcitationenergyandDTM ...............60 4-6Oscillatorstrengthconsistencycheck .......................63 4-7MolecularorbitalsofwaterHF/WMR ........................67 6-1Dissociationenergyforhydrogenuoride .....................80 6-2NPEforhydrogenuoride ..............................80 6-3Orbitalspaceandperformanceforhydrogenuoride ...............89 6-4Dissociationenergyformolecularuorine .....................90 6-5NPEformolecularuorine ..............................91 6-6Orbitalspaceandperformanceformolecularuorine ...............93 6-7Optimizedgeometryofethylene ..........................98 6-8Barrierofrotationforethylene ............................99 6-9Selectbondlengthsofbicyclobutaneisomers ..................102 6-10Energeticsofbicyclobutaneisomerization .....................104 6-11Vibrationalfrequenciesofbicyclobutaneisomerizat ion ..............105 7

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LISTOFFIGURES Figure page 1-1Electronicstructurediagram .............................14 1-2MBPT(2)Hilbertspace ...............................15 1-3Scalingandcorrelationrecoveryofpost-Hartree-Fock methods .........16 2-1VisualizationoftheFrsterCycle ..........................23 2-2Structuresofphenol, -naphthol,and -naphthol. ................23 2-3TheatomiclabelscorrespondingtoTable2-1. ...................27 2-4Hatree-Fockorbitalsoffree-phenolandphenol-ammoni aclustercorresponding totheexcitationslistedinTable2-2 .........................27 2-5VerticalexcitationspectrausingtheCIS,TD-DFT/B3LY P,EOM-CCSD,and STEOM-CCSDmethodslistedinTable2-2 ....................32 2-6TheatomiclabelscorrespondingtoTable2-4. ...................35 2-7Hatree-Fockorbitalsof -naphtholcorrespondingtotheexcitationslistedin Table2-5. .......................................38 2-8Hatree-Fockorbitalsof -naphtholcorrespondingtotheexcitationslistedin Table2-5. .......................................39 4-1VerticalexcitationspectrumofNaOH ........................53 4-2EnergiesandDTMofNaOHalongthedissociativecoordina te .........56 4-3Zeroth-ordercalculatedabsorptioncrosssectionofNa OH ............58 4-4EnergiesandDTMofNaOHalongthebendingcoordinate ............61 4-5Vibrationalwavefunctioninapotential .......................62 4-6Convolutionofcrosssections ............................63 4-7FinalabsorptioncrosssectionforNaOH ......................64 4-8ExperimentalandcalculatedVUVabsorptioncrosssecti onforwater ......66 4-91 A 2absorptioncrosssectionofwater .......................68 5-1Activespacediagram ................................71 6-1DissociativePESenergyerrorwithrespecttoCCSDTofFH ,(1) .........79 6-2DissociativePESenergyerrorwithrespecttoCCSDTofFH ,(2) .........81 8

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6-3DissociativePESenergyerrorwithrespecttoCCSDTofFH ,(3) .........82 6-4AmplitudesfromthedissociativePEScalculationsFH ..............83 6-5GraphicaloutputfromASDAforFH .........................86 6-6DissociativePESenergyerrorwithrespecttoCCSDTofu orinemolecule ..92 6-7GraphicaloutputfromASDAforuorinemolecule ................93 6-8Isomerizationofbicyclobutanestructures .....................100 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DEFLATINGTHEDIMENSIONALITYCURSEINELECTRONICSTRUCTUR E CALCULATIONS By AnnMelnichuk May2013 Chair:RodneyJ.BartlettMajor:Chemistry Many-bodyproblemswhetherclassicalorquantumsufferfro mwhatisknownas a dimensionalitycurse .Insolvingphysicalproblems,wearegenerallyinterested in quantifyinginteractionsbetweenphysicalentitiesandas thequantityoftheseentities increases,thenumberofinteractionequationsonemustsol veincreasesmuchfaster. Theworkdescribedinthisdissertationpresentsseveralso lutionsforthereduction ofthedimensionofcalculationsofelectronicstructureof molecules.Techniquesand examplesaredrawnfromcalculationsofexcitedstates,UV/ Visabsorptioncross sections,andhigh-leveltheoreticaltreatmentofmulti-r eferenceelectronicstructure problems. Thetechniquesdescribedheremaybecombinedtoachievehig hlyaccurate theoreticalresultsforchallengingexamplesatonlyasmal lfractionofthecost. Speed-upupwardsof500xwereobservedforseveralsystemsv ersustheestablished methodsandseveralexamplesarepresentedwhichwouldnotb eotherwisepracticalto dogiventhecurrentcomputationallimitations. 10

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CHAPTER1 INTRODUCTIONTODIMENSIONALITYISSUESINQUANTUMMECHANIC S Inelectronicstructuretheory,tocalculatethecorrelati onenergybetweenelectrons isaproblemwhichsuffersfromaso-calleddimensionalityc urseasdomostn-particle problemsinphysics.Thesimplestapproximationtoelectro ncorrelationenergyisa perturbativecorrectionofthesumofallpairofelectronsi nteractionstoareference energywhichisobtainedbyavariationalminimizationcalc ulationforeachelectronina eldofn1electrons(Hatree-Fockmethod). 1.1Hatree-Fockreference SeveralapproximationsaremadetothefullSchrdingerEqu ation,^ H ( r t )( r t ) = @ @t ( r t )tomakeitapplicabletoamolecularsystem.Initially,theH amiltonian(^ H)and thewavefunction()areseparatedintoafunctionoftimeandafunctionofspace .The calculationofthegroundstateenergywithHartree-Fockin volvesonlythespatialportion oftheSchrdingerEquation:^ H = PA 1 2 m Ar2 ANuclearkineticenergy+PB
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setofGaussianfunctionscalledabasisset.Theatomicbasi ssetfunctionsdescribe theatomicelectronicstructuretovariousdegreesofaccur acyandareuniquetoeach atom.AftertheHatree-Fockequationsaresolvedandthemin imumoftheenergyis obtained,thelinearcombinationofatomicorbitals(LCAO) representthemolecular orbitalelectronicstructureofthegroundstatereference TheHatree-FockcalculationnominallyscalesasO ( N 4 )whereNisthenumberof basisfunctionsduetothecomputationof2-electronintegr alquantities.Theoverheadis greaterduetotheneeddocallaneigensolverateachiterati onforamatrixsizeofN 2. TherearelinearlyscalingHartree-Fockapproachesbutthe yarenotusedinthisstudy sincecalculationofthereferenceisneverthetimelimitin gstephere. ThegroundstatereferenceisideallyasingleSlaterdeterm inant( j 0i=j 12 ...Ni )where isamolecularorbital.IntheunrestrictedHartree-Fock(U HF) calculationthe and electronsaretreatedasseparateentitieswhichmayleadto abreakingofspinandspatialsymmetryaswellasalowergrou ndstateenergythan theenergyfromarestrictedHartree-Fockcalculation(RHF )where and electrons areindistinguishable.Inawell-behavedsinglereference system,theUHFandRHF solutionsarevirtuallyidenticalbutinmulti-references ystems(wheremorethanone Slaterdeterminantisneededtodescribetheelectronicsta te),theUHFwavefunction canbedramaticallydifferentthantheRHFwavefunction. Inthiswork,preferenceisgiventotheRHFwavefunctionfor closed-shellmolecules becauseitisaneigenfunctionofthespinoperatorwhichlea dstowholevaluesforthe spinquantumnumberwhereastheUHFwavefunctionmayleadto acontaminatedspin quantumnumbersuchas1.2,2.4,etc.Inahigh-spinopen-she llmoleculesuchasa hydroxyradical,theUHFwavefunctioniswellbehavedneart heequilibriumgeometry andproduces 2forthespinnumberuptocomputationalaccuracy.Restrict ed open-shellHatree-Fock(ROHF)referencemaybeusedtostri ctlyimposethespin eigenfunctionpropertyoftheopenshellwavefunction.For difcultmulti-referencecases 12

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whichalsohappentobeopen-shelltheROHFsolutiontendsto haveconvergence difcultiessuchasoscillatingbetweentwonearlydegener atesolutions. TheRHFreferencedoesnotincludeanyelectroncorrelation energy.Eventhough theelectroncorrelationenergyisaverysmallportionofth etotalenergy,thecorrelation energyofanelectronpairis 20 kcal=molwhichisontheorderofchemically-relevant quantities.Withoutcorrelation, abinitio calculationstendtolosetouchwithreality. Therearetwomaintypesofcorrelation:dynamicandstatic. Thedynamiccorrelation manifestsitselfinalargenumberofsmallcontributionsfr omSlaterdeterminantsother thanthegroundstate j 0i .Thestaticcorrelationmanifestsitselfincaseswherethe re aretwoormoreSlaterdeterminantswhichhaveasignicantc ontributionofwhich j 0i onlydescribesapartofthetotalgroundstatewavefunction .Post-Hartree-Fock methodswhichaccountforthecontributionsofotherdeterm inantsareneededtoobtain experimentally-relevantresults.Thecalculationofelec troncorrelationiswhereone becomesfacedwithadimensionalexplosionoftheproblem. 1.2Dimensionalityissuesinelectronicstructure 1.2.1Groundstate Forthebenetofclarityandsimplicity,thefollowingassu mptionsaremadeinthe discussionthatfollows:1.Thesystemisclosedshellwithanevennumberofelectrons ;referredtoasRHF (restrictedHartree-Fock). 2.TwoindistinguishableelectronsoccupyeachHFenergyle vel. 3.Thephysicalspacedescribinganelectronwithrespectto theotherelectrons generatesasetofone-andtwo-electronintegrals. Byassumption(2),weonlyneedtocalculateenergycontribu tionfromoneofthe electronsinanenergylevelandsimplydoubleit. Theintegralsarecalculatedpriortothecorrelationenerg ycalculation.Theyare storedasatensorofrank4andhavedimensionofo 2 v 2.Theindexo(numberof 13

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Figure1-1.Diagramshowinganexampleofatypicalelectron icstructureproblem occupiedorbitals)isafunctionofhowmanyelectronsarein thesystem,o = ( Num e )=2andtheindexvisthenumberofvirtualorbitals.Graphically,thepartiti oningoftheoandvspaceisrepresentedinFigure 1-1 .EachlevelinFigure 1-1 denotesalledoran emptyorbital.Theenergyvaluesofthelevelsaretheeigenv aluesoftheRHFequations andtheorbitalsaretheassociatedeigenvectors. Inordertoincludeelectroncorrelation,westartwiththeu nperturbedsystemas showninFigure 1-1 andallowtheelectronstomovefromtheoccupiedorbitalsto theunoccupiedorbitals.Thisallowsforsamplingofdeterm inantsotherthanthe j 0i determinant. ByvirtueofBrillouin'sTheorem,determinantswhichdiffe rfrom j 0i bymovinga singleelectronfromtheoccupiedspacetothevirtualspace donotdirectlycontribute tothecorrelationenergybecausethematrixelement h 0j Hj a ii is0.Asarstorder approximationofthemanybodyperturbationtheory(MBPT2) ,weonlyallowfor twoelectronstomoveatatime;makingasetofdoubly-excite dstatesknownasa doubly-excitedHilbertspace. Figure 1-2 showssomeofthecorrelationenergycontributionswhichar isefrom movingapairofelectronsintheorbitalspace.Theindexnot ationisE 2( A I B J )whereIistheindexoftheorbitaloforiginofoneoftheelectronsan dAistheindexofits 14

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Figure1-2.DiagramshowingseveralelementsoftheHilbert space(onepaironly) startingfromthezerothorderwavefunction(RHF).Therear eo 2 v 2 + 1elementsinthisspace. destinationorbital.IndexJisoftheorbitaloforiginoftheotherelectronandindexBis ofitsdestinationorbital. Itisclearthatevenforarstorderapproximationtothecor relationenergythe numberofcontributingelementscangetverylargeanditwil lbeevenlessmanageable, asweallowformorethantwo-particleinteraction. ThefollowingformulaisusedtocomputetheE 2contributionsandthenalvalueforE 2:E 2( A I B J ) = 2 G ( A I B J ) 2G ( A I B J )G ( B I A J ) ( I ) +( J ) ( A ) ( B )(1–1)E 2 = bfXA = o +1 oXI =1 bfXb = o +1 oXJ =1 E 2( A I B J )(1–2) whereGisthe2-electronintegraltensorand isavectoroforbitalenergies.Thetotal energyforthesystemisthenpresentedasE = E RHF + E 2. ThemaindrawbackoftheMBPTntheoryistheslowconvergence torecover 100%ofthecorrelationenergycombinedwithasteeplyclimb ingcost:MBPT2 (O ( o 2 v 2 )),MBPT3(O ( o 2 v 4 )),MBPT4(O ( o 3 v 4 )).Thebestsolutiontoobtainthe highestpercentageofcorrelationenergyforthelowestcos tisthecoupledcluster(CC) theory[ 3 4 8 – 10 27 28 84 123 131 ].TheCCmethodshavetheadvantageofbeing 15

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Figure1-3.Scalingandcorrelationrecoveryofselectpost -Hartree-Fockmethods adaptedfromBartlettandMusialJCP2007[ 113 ] size-extensiveandinnitetoordersofMBPTwhichleadstoa muchfasterconvergence to100%ofthecorrelationenergy[ 113 ]asshowninFigure 1-3 IntheCCansatztheotherdeterminantsareaccessed via a^ Toperatorwhichis composedofcreationandannihilationoperators y and .Thefollowing^ T 2-operator createsthedeterminantspaceshowninFigure 1-2 :^ T 2 = 1 4Xa b i j t ab ij ^i ^j ^ ya ^ yb(1–3) wheret ab ij(oneofmanytypesoft-amplitudes)arethesolutionstotheCCequations below.TheentireCCwavefunctionconsistsofallordersoft he^ T-operator: j CCi= e ^ Tj 0i (1–4) 16

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e ^ T1 + ^ T 1 + ^ T 2 + 1 2! ^ T 2 1 + ^ T 3 + ^ T 1 ^ T 2 + ...(1–5) Specically,thepopularsinglesanddoublescoupledclust ermethod(CCSD)is: j CCSDi= e ^ T 1 + ^ T 2j 0i (1–6) whichscalesasiO ( o 2 v 4 )andtheCCSDTmethod j CCSDTi= e ^ T 1 + ^ T 2 + ^ T 3j 0i (1–7) whichscalesasiO ( o 3 v 5 ).Sincebothareiterativemethods,iisthenumberofiterations neededforconvergenceofthet-amplitudes.ForexampleintheCCSDmethod, convergenceisreachedwhenconditions h a ij^ He ^ T 1 + ^ T 2j 0i= 0and h ab ijj^ He ^ T 1 + ^ T 2j 0i= 0aresatisedbysolvingasetofnon-linearequations. Thecorrelationenergyiscomputedonceconvergenceisreac hed.TheCC energyisexpressedbysubstitutingasimilarity-transfor medHamiltonian( H)intothe SchrdingerEquation:e^ T ^ He ^ Tj 0i= Hj 0i= E CCj 0i (1–8) h 0j Hj 0i= E CCh 0j 0i (1–9) h 0j Hj 0i= E CC(1–10) Equation 1–10 istrueforallordersoftheCCtheorybecausehigherorder^ T-operatorsdonotmakeacontributiontoenergysincetherea reatmost2-particle termsintheHamiltonian( cf. Baker ASCampbell ASHausdorffcommutatorrelation). TheCCmethoddescribedaboveisbasedonasinglereference j 0i uponwhich allthecontributionsfromotherreferencesarecomputed via theexponentialansatz. Inthecasewhereaproblemisdominatedbydynamiccorrelati on,theCCSDand thesubsequentCCSD(T)(CCSDwithanon-iterativecontribu tionfromthreeterms 17

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fromtheCCSDT)canrecoverover99%ofthecorrelationenerg y.However,dueto theperturbativenatureoftheCCequations,casesofstatic correlationwhenseveral referencescontributegreatlytoasinglestate,oneneedst ogobeyondCCSDto obtainacorrectdescriptionoftheproblem.Insuchcasesth eCCcalculationsbecome unmanageablyexpensive.Anynon-iterativemethodsbasedo nasinglereferencetend tofaildramaticallyformulti-referenceproblemsaswell. Aclassicexampleofthisisthe ozonemoleculewhichhasfourmajordeterminantsinthegrou ndstateatequilibrium.In ordertogetquantitativelycorrectvibrationalfrequenci es,oneneedstosolveCCSDTQ equations. Howtousesingle-referenceCCtheorytotacklecaseswheres tatic and dynamic typesofcorrelationarepresentwillbethesubjectofChapt ers5and6ofthisdissertation. 1.2.2Excitedstate Inthiswork,verticalexcitationenergiesarecomputedusi ngconguration interactionwithsingles(CIS)[ 46 ],equation-of-motionsinglesanddoubles(EOM-CCSD)[ 148 ] methodsandthesimilaritytransformedequation-of-motio n(STEOM-CCSD)method[ 6 ] tocomputeexcitedstateenergies.Thecomputationofexcit edstatessuffersfrom asimilardimensionalexplosionproblemasthegroundstate CCtheory.Chapter2 describesseveralexamplesregardingtheaccuracyofthese methods. TheCISmethodistheleastexpensivesinceonlysingly-exci teddeterminantsare includedfor.Inessence,theCISmethodndshigh-energyso lutionstotheHartree-Fock equationsandassuchdoesnotincludeanyexplicitelectron correlation.TheCIS excitationenergyforak thtargetexcitedstateis: !k = E a i ( CIS ) = E 0 ( HF ) i +ahiajjiai (1–11) whereicanbeanyoccupiedmolecularorbitalandacanbeanyvirtualmolecularorbital. WhileCISdoesnotaccountforelectroncorrelation,thewav efunctiongeneratedbyCIS isoftenusedasastartingguessforsolvingtheEOM-CCequat ions. 18

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TheEOM-CCmethodcanbeusedtocomputetheenergyandproper tiesofak thtargetexcitedstatewhetheritbeelectronattachment(EAEOM),ionizationpotential (IP-EOM)orexcitation(EE-EOM)withelectroncorrelation .Itappliesalinearexcitation operator(^ R k)onagroundstatecoupledclusterwavefunction:^ H ^ R k e ^ Tj 0i= E k ^ R k e ^ Tj 0i (1–12) orintermsof H: H ^ R kj 0i= E k ^ R kj 0i (1–13) where^ R kis:^ R k = ^ R 0 + ^ R 1 + ^ R 2 + ... = ^ R 0 +Pi a r a i ^i ^ ya +Pj
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treatmentofCTstates2)indirectinclusionofsometriples 3)capabilitytosolvefor manyk-stateswithvirtuallynooverhead.Themaindrawbackisthe dependenceonthe completenessoftheactivespace.Chapter2showsafewexamp lesofCTstateswhere STEOM-CCSDissuperiortoEOM-CCSDinaccuracyandChapter5 addresseswaysto automaticallyselectactivespaceswhichcanbeapplicable toaSTEOM-CCcalculation. ThestepstocalculateSTEOM-CCSDexcitationenergiesare( adaptedfrom[ 6 ]): SolvethegroundstateCCSDequations Store Honthedisk ChooseactiveoccupiedorbitalsandsolvetheIP-EOM-CCSDe quations.Storethe resulting^ R k(IP)inan^ S -operator. ChooseactivevirtualorbitalsandsolvetheEA-EOM-CCSDeq uations.Storethe resulting^ R k(EA)inan^ S +-operator. Performasimilaritytransformon H:^ G = e( ^ S+ ^ S + ) He ( ^ S+ ^ S + ) Diagonalize^ Goverasetof^i ^ yaj 0i congurationswhereorbitalsiandabelong totheactivespace. TheSTEOM-CCSDcalculationhasscalingofN IP iO ( o 2 v 3 ) + N EA iO ( ov 4 ) + jO ( ov ), thelattertermbeingthediagonalizationof^ G-matrixstep.SincebothEAandIPsteps nominallyscaleasO ( N 5 )andtheEEscalesasO ( N 6 ),STEOMisguaranteedtobe fasterifseveralexcitedstatesarerequestedandthecompu tationaldisparitybetween EOMandSTEOMgrowsasthenumberofEOM-CCSDrootsbecomesla rger. 1.3Dimensionalityissuesinmolecularstructure Therstexampleofthisisincalculatingthevibronicenerg iesofmolecules.Any non-linearmoleculehas3 N6vibrationaldegreesoffreedomwhereNisthenumber ofatomsinthemolecule.Fromeachvibrationalleveloftheg roundstate,onecangetto anysymmetry-allowedvibrationallevelsoftheexcitedsta te,thuscreatingthespectral linebroadeningseeninUV/Visspectroscopy.Forlargemole cules,calculationofallthe possiblecouplingsofvibronicstatescanbecomputational lyprohibitiveanddoingsofor 20

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dissociativeexcitedstatesprecludestheuseofthequadra ticpotentialenergysurface (PES)approximationswhichmakethesecalculationsatallf easible. Inthiswork,amethodisintroducedwhichanalysesthegroun dstatevibrational energylevelstodeterminetheirlevelofcoupling.Theleve lswhichcouplestrongly arepre-selectedtogotothenextstepintheprocess.Analyt icaltsaremadeforthe pre-selectedvibrationalmodesforthegroundstateandexc itedstatePESaswellas thedipolestrengthsurface.Vibronicexcitationcrosssec tionisthencomputedusing discretevariablerepresentation(DVR).Temperature-dep endentbroadeningcanbe calculatedusingBoltzmann-weightedaveragingoftheabso rptioncrosssectionsfrom eachofthegroundstatevibrationalenergylevels. Thecombinationofcouplingpre-screeningandanalytical tsofthesurfacesmakes thisapproachscaleas nwhere isthetimeittakestocomputetheenergysurface(to bedonewithwhateverleveloftheorynecessaryforgoodresu lts)andnisthenumberof pre-screeneddegreesoffreedomforthefunctionalgroupre levanttothestudy. 1.4Implementationtechniquesandsoftwareused InthisworkbothserialACESII[ 147 ]andtheparallelACESIII[ 80 92 93 ]are used.Theauthor'smodicationandimplementationofactiv e-spaceCCmethodsare donewithinACESII. Thediscretevariablerepresentation(DVR)method[ 90 ]isadaptedtovibronic spectroscopymethodsandanoriginalstand-aloneprogrami swritteninF90withan interfacetotheACESIIprogram. Anautomaticactivespaceselectionalgorithmiswrittenus ingtheRscripting languagewithaninterfacetotheACESIIprogram. 21

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CHAPTER2 CHARGETRANSFEREXCITEDSTATES 2.1Background Chemicalreactionsthattakeplaceonexcitedstatepotenti alenergysurfacesare thesubjectofmanyexperimentalandtheoreticalstudies,s incereactivitymightbe greatlyenhancedbytheelectronicexcitationofoneormore ofthesubstrates.One classofthesereactionsisexcitedstateprotontransfer(E SPT)whichissummarizedby theFrsterCycle[ 47 ]showninFigure 2-1 ThecorrectdescriptionofCTisofinterestinseveralstudi es.Questionsaddressed includetheassessmentofthedegreeofCTintherstexcited state,assumedto besignicantby[ 44 ],aswellasinotherpotentialCTstates.Also,sinceitiskn own thatTD-DFTcannotdescribesuchstatescorrectly,moresop histicatedmethodslike EOM-CCandSTEOM-CCareappliedhere. 2.2Backgroundonexamples ThefreemoleculesareshowninFigure 2-2 .Theammoniacomplexisformedby creatingahydrogenbondbetweenthenitrogenandtheproton onthehydroxygroup. Onlythe cis isomerofthe -naphtholisconsidered. Previoustheoreticalstudiesofphenol-ammoniaclustersh avebeencarried outforthegroundstate,therst excitedstateandthe excited state[ 32 141 142 ].Thersttwophenolexcitedstateswerealsocharacterize dat theCASSCFleveloftheory[ 51 ].Experimentalstudiesofphenolandphenolammonia clustersareabundantintheliterature[ 65 68 108 119 127 143 154 ].The -naphthol ammoniaclustershavebeenstudiedwithresonance-enhance dmultiphotonionization (REMPI),resonanttwo-photonionization(R2PI),laserind uceduorescence(LIF),and uorescenceemissionspectroscopy[ 24 25 56 58 63 72 150 ]withsupportfrom ab inito study[ 56 150 ].The -naphtholammoniaclustershavebeenstudiedwithREMPI, R2PI,uorescenceemissionspectroscopy,andwithUVabsor ptionandemissioninan 22

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6 6 ROH ROH RO +H + RO *+H + E E 0 Figure2-1.VisualizationoftheFrsterCyclewhereROHist hephotoacid.Inthiswork theRgroupsconsideredarebenzyl, -naphthyl,or -naphthyl. E0 isshown tobered-shiftedfrom E. OH OH OH Phenol aNaphthol bNaphthol Figure2-2.Structuresofphenol, -naphthol,and -naphthol. Argonmatrix[ 19 36 44 67 104 129 ];andinsolution[ 144 ].Forfurtherinformation onspectroscopictechniquesusedtostudysolvationred-sh iftsinthegasphaseconsult Dissent etal. [ 34 ].ForareviewofESPTofphenolandnaphtholphotoacidsseet he followingreferences[ 1 33 73 85 153 166 ]. Recentlyavailablehighresolutiongasphasespectraof -naphtholandthe -naphthol-ammoniacomplexshowasolvatochromicredshift of585cm -1 intheenergy ofuorescenceoftherstexcitedstate.Fleisher etal. explaintheobservedred-shiftby achargetransfer(CT)betweenthesoluteandthesolvent.Th eyanalyzethedifference betweentheEDMofthegroundstateandtheexcitedstateofth ecomplex,compared 23

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tothedifferencebetweentheEDMofthegroundstateandthee xcitedstateofthefree molecule. BydecomposingtheEDMintofourcomponents:thedipolemome ntsofthe naphthol,theammonia,andtheinduceddipolemoment;thedi fferencebetweenthe totaldipolemomentofthecomplexandthesumofthefourcomp onentslistedwas consideredthedipolemomentofchargetransfer.Suchanemp iricalmodelneglects severalfeaturesthatpredictivelevelsof abintio theorycanresolve.Theargumentfor chargetransferbeingresponsibleforthered-shiftofexci tedstatesofhydrogenbonded speciesisanoldone,datingbacktoNagakuraandGouterman( 1957)forphenol, -naphthol,and -naphtholspecically[ 117 ].Thewavefunction-baseddescription andclassicationofCTdonorsandacceptorsdatesbacktoMu lliken(1952)whose methodsarewidelyaccepted[ 112 ].However,theconclusionofFleisher etal. gas phaseexperimentisthatCTisresponsibleforthered-shift oftherstexcitedstatewhile theconclusionofSolntsev etal. incondensedphaseexperimentsisthatHydrogen bondinginteractionsareresponsibleforthered-shiftoft herstexcitedstate.Inthis paperweaddressthisdiscrepancy.Whileitisacceptedthat the transitions arechargetransferthediscussionoftheCTcharacterof transitionscontinues despiteearlyCNDOcalculationssuggestingthatitisnotth ecase[ 13 ]. Asimilartheoreticalstudyisdoneforthe isomerofnaphtholsinceitisknownthat thisisomerislessphoto-acidicthanthe isomer[ 73 117 ]andthereforemaydisplay differentexcitedstateproperties.Therehavebeensevera lpreviousDFT,CIS,CASSCF andCASPT2studiesofthephenolexcitedstatedbutnoEOM-CC SDorSTEOM-CCSD studyhavebeenperformed.Thecoupledclustermethodsshou ldprovidemoredenitive resultsthanthepreviousmethodscited.Mostoftheexperim entalliteraturecitedhere reportsthe0-0vibronictransitionbutnottheverticalabs orption(oremission)thus makingadirectcomparisontotheoreticalvaluesdifcult. Fortunately,theexcitation energyshiftsduetothepresenceoftheammoniasolventmole culesappeartobelittle 24

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affectedwhethertheshiftiscalculatedfromthe0-0vibron icbandorfromsomeother band(5-25wavenumbererror)[ 25 36 ]soacomparisonoftheshiftsbetweentheory andexperimentisreasonable.Furthermore,inanumberofex perimentalstudiesitis notedthatthe0-0peakisthehighestintensitypeakthusmak ingitaverticaltransition fortherstexcitedstate.Sincetherstexcitedstateofth earomaticcompoundsinthis studyisexpectedtobea transition,wedonotexpectabiggeometrychange betweenthegroundandtheexcitedstate,meaningthattheve rticalexcitationenergy willnotberadicallydifferentthanthe0-0bandenergy. 2.3Methods Webeginwithphenolanditsammoniacomplex.Phenolactsasa photoacidina similarwayasnaphthol(seethereviewofDavid etal. [ 33 ]).Thissystemisaccessible toalloftheexcitedstatecapabilitiesofACESIIwithoutco mpromisingtheessenceof theproblem.Thegroundstategeometriesareoptimizedusin gCCSDwithaDZPbasis whoseexponentsofpolarizationfunctionsaretakenfromco rrelatedcalculations[ 134 ]. Verticalexcitationenergiesarecomputedusingcongurat ioninteractionwithsingles (CIS)[ 46 ],EOM-CCSDandSTEOM-CCSDmethodstocomputeexcitedstate energies. Forcomparison,aTD-DFT[ 48 ]calculationusingahybridB3LYP[ 12 ]functional.All DFTcalculationsareperformedusingGAMESS[ 137 ]software.ADZPbasismodied withdiffusefunctionsisusedinexcitedstatecalculation sinordertoproperlydescribe anyRydbergstates. Thegroundstategeometriesof -naphtholand -naphtholandtheirammonia complexesareoptimizedusingMBPT(2)[ 7 55 ]witha6-31++G**basisset[ 54 ]. Singlepointenergycalculationsareperformedontheoptim izedgeometrieswith CCSD/6-31++G**andwithCCSD(T)/6-31++G**[ 11 ].Geometryoptimizationandsingle pointenergycalculationsaredonewithACESIII.Excitatio nenergiesarecomputed usingCISandEOM-CCSDmethods. 25

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ArestrictedHartee-Fock(RHF)referenceisusedforallcou pledclustercalculations. ElectricdipolemomentsarecalculatedwithEOM-CCSDandST EOM-CCSDinACESII. Allparallelgeometryoptimizationcalculationsareaccom plishedontheUniversityof FloridaHighPerformanceComputingCenterwithAMDOpteron (2.2GHz)nodes.The resourcerequestsaremodest(24-64processors)sothatthe performanceofACESIII maybeascertainedinanaverageworkingenvironmenteventh oughithasbeenshown thatACESIIIscalesexceptionallywellontens-of-thousan dsofprocessors[ 93 ].The excitedstateparallelcalculationsareaccomplishedonCr ayXE6AMDOpteron(2.3 GHz)nodesattheArcticSupercomputerCenterwith256-512p rocessors.Theorbital plotsaredoneusingtheMacMolPltsoftwarepackage[ 16 ]. 2.4Resultsanddiscussion 2.4.1 Abinitio spectrumofphenol Theexcitedstateenergyandpropertiesaredeterminedfort hefreephenol moleculeandthephenol-ammoniacomplex.Thenatureofthe rstfourexcitedstatesis determinedfromorbitalanalysisandfromtheelectricdipo lemoment.Thedifferencesin energiesandpropertiesduetothesolvationofphenolbyasi nglesolventmolecule(in thiscaseammonia)aredocumented.Thegroundstategeometr iesareoptimizedatthe CCSD/DZPlevelunlessstatedotherwise.Theoptimizedgeom etricparametersforthe freephenolandthephenol-ammoniacomplexarelistedinTab le 2-1 andthelabelsare showninFigure 2-3 Theequilibriumgeometriesareingoodagreementwiththeex perimentalvalues[ 81 ] aswellaswiththeprevioustheoreticalstudies[ 32 51 142 ].TheCCSD/DZPmethod yieldsasmallerHydrogenbondlength(1.8912)asopposedt otheCASSCFresults: 2.042(SobolewskiandDomcke)and2.07(Daigoku etal. ). Theresultsoftheexcitedstatecalculationsarepresented inTable 2-2 .See Figure 2-4 fortheplotsoftheorbitals.Thesuperscriptxsigniesthattheexcitedstate andtheorbitalsarethoseofthecomplex.Theorbitalswitht hehighestcoefcientsfor 26

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H10 H11 C6 C5 C4 H9 H8 H7 C3 C2 C1O12 H13 N14 H15 H16 H17 Figure2-3.TheatomiclabelscorrespondingtoTable 2-1 Figure2-4.Hatree-Fockorbitalsoffree-phenolandphenol -ammoniacluster correspondingtotheexcitationslistedinTable 2-2 .Thesuperscriptxsigniesthattheexcitedstateandtheorbitalsarethoseof thecomplex.The designationstandsfora valencetransitionandthedesignationR standsforaRydbergstate. 27

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Table2-1.Optimizedgeometryparametersforfreephenolan dphenol-ammonia complexwithCCSD/DZP.ThedistancesareinAngstromandthe anglesare indegrees. CoordinatePhenolPhenol-NH 3 C 1 C 21.40541.4094C 2 C 31.40611.4052C 3 C 41.40311.4037C 4 C 51.40731.4070C 5 C 61.40191.4016C 2 H 71.09271.0909C 3 H 81.09091.0915C 4 H 91.09001.0901C 5 H 101.09101.0915C 6 H 111.08981.0901C 1 O 121.37071.3583O 12 H 130.96220.9775H 13 N 14-1.8912N 14 H 15-1.0195N 14 H 16-1.0195N 14 H 17-1.0195 \C 1 C 2 C 3119.7120.0 \C 2 C 3 C 4120.6120.8 \C 3 C 4 C 5119.2119.0 \C 4 C 5 C 6120.8120.8 \H 7 C 2 C 1120.0119.8 \H 8 C 3 C 2119.3119.3 \H 9 C 4 C 3120.4120.5 \H 10 C 5 C 6119.3119.3 \H 11 C 6 C 1119.0118.7 \O 12 C 1 C 6117.1117.6 \H 13 O 12 C 1108.1109.6 \N 14 H 13 O 12-170.5 \H 15 N 14 H 13-106.9 \H 16 N 14 H 15-105.7 \H 17 N 14 H 15-105.7 ]H 16 N 14 H 15 H 13--124.0 ]H 17 N 14 H 15 H 16--112.0 28

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Table2-2.TheexcitationenergyisreportedineV.Theoscil latorstrength(f)isunit-less.ThevaluesforEDMare reportedinDebye.ThegroundstateEDMforphenolis1.32Dan dthegroundstateEDMforphenol-NH 3 is 3.97Dforcorrelatedmethods.Thegroundstater 2forphenolis705.9andthegroundstater 2forphenol-NH 3 is 1469.4.Theexcitedstatelabelsandorbitalscorrespondto Figure 2-4 .Allcalculationsareperformedatthe optimizedgroundstategeometryattheCCSD/DZPlevelwithD ZP++basisset.Thedesignationstandsfora valencetransitionandthedesignationRstandsforaRydber gstate.Thesuperscriptxsigniesthatthe excitedstateandtheorbitalsarethoseofthecomplex. StateB3LYPCISEOM-CCSDSTEOM-CCSD E(eV)fE(eV)fE(eV)f EDM(D) r 2E(eV)f EDM(D) r 2 Phenol 14.950.0315.810.0474.900.0210.070.64.410.0180.130.5R 15.210.0006.280.0015.660.0009.4138.95.570.0009.2739. 0R 25.650.0046.560.0176.180.0057.1943.56.110.0007.3644. 3 25.710.0326.080.0026.230.0490.114.66.060.0790.754.0Ph-NH 3 x 14.840.0385.730.0614.830.0250.44-0.14.340.0240.56-0. 2R x 14.390.0006.100.0025.480.0005.1956.95.370.0005.3657. 7R x 25.260.0036.340.0175.850.0065.2056.55.760.0065.6156. 3 x 25.580.0536.030.0076.040.0700.077.65.830.1211.076.0 29

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eachexcitedstateareusedtoidentifythestateslistedinF igure 2-4 andTable 2-2 ACTexcitedstatewouldbeexpectedwheninthepairoforbita lswiththedominant contributiontothetransitionmatrix,onewouldbeontheso luteandoneonthesolvent. DependingonthedirectionofCT,suchstateswouldbedomina tedbyatransitionfrom eitheranoccupiedorbitalonthephenoltoavirtualorbital onthesolvent(suchasR x 1)or fromanoccupiedorbitalonthesolventtoavirtualorbitalo nthephenol.R x 1istherst statewhichisobservedtoinvolveatransitionbetweenthep henoloccupiedorbitaland avirtualorbitalontheammonia.ThisisaRydbergstateandd uetoaminimalorbital overlapdoesnothaveasignicantoscillatorstrength. ThemaindifferenceintheTD-DFTspectraistheunderestima tionofthevertical excitationenergiesaswellasthesignicantloweringofth eexcitationenergyoftheCT state(R x 1)comparedtothewavefunctionspectra.TheCISenergiesare signicantly higherthantheEOM-CCSDandSTEOM-CCSDresultsandtheosci llatorstrengthsofR 2,R x 2, 2and x 2statesarereversed.ThisiswellillustratedinFigure 2-5 whichshows thelinespectraforseveralofthemethodslistedinTable 2-2 ascomparedtothersttwo statesfromKimuraandNagakura[ 69 ].Theexperimentalvaluesforexcitation energiesandoscillatorstrengthfor 1and 2stateswere4.6eV(f=0.020)and5.77 eV(f=0.132)respectively.Theexperimentaldatapointsarett edwithaGaussian lineshapeforbettervisualization.Sincethiswasaconden sedphasespectrum,the agreementbetweencalculatedvaluesandexperimentalvalu eisnotexpectedtobe perfect,butitshouldbeatleastqualitativelyrightsince theheptanesolventisnotpolar andshouldnotaddnoticeablesolvationeffects.Thebestag reementisshownbythe STEOM-CCSDmethodwhichgetsbothenergiesandoscillators trengthsqualitatively correct.TheratioofoscillatorstrengthintheEOM-CCSDsp ectrumbetween 1and 2statesislowerthanexperimentbuttheenergiesarestillon target.TheCISspectrum getsneithertheenergiesnortheoscillatorstrengthsqual itativelycorrectandwhile 30

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TD-DFT/B3LYPenergiesarecertainlybetterthanCIS,theos cillatorstrengthsarenot consistentwithexperiment. Theexcitationenergyandoscillatorstrengthfor 1isconsistentwithseveral CASSCFandCASPT2calculations[ 32 51 142 ].However,theR 1whilebeing consistentwiththeCASPT2calculationsofSobolewskiandD omckeaswellas Daigoku etal. doesnotmatchthesecondexcitedstatefoundbytheCASPT2 calculationofGranucci etal. .Thecorrelatedmethodsmatchtheenergyandtheorbital assignmentsoftheCASPT2resultsofSobolewskiandDomckea nd(Daigoku etal. ): 5.77eV(5.81eV)and5.47eV(5.76eV)forR 1andR x 1respectively.However,fortheR 1state,thecorrelatedmethodsdonotmatchtheexcitationen ergiesoftheCASPT2 resultsofGranucci etal. ,whichis6.26eVandcloserinenergytoeithertheR 2orthe 2state.Uponfurtherinvestigation,thesecondexcitedstat einGranucci etal. isthe 2state,whichisthesecondexcitedstateinA00 symmetry,butnotthesecondstate overall.Theconfusionwascausedbyamistakeinthesymmetr yassignmentsofexcited statesinTables2and3inthepaperofGranucci etal. :therstandsecondstateswere assignedtoaBandAsymmetryrespectivelyinTable2andbothwereassignedtoBsymmetryinTable3.Afterthecorrectedassignmentismade, theagreementbetween theEOM-CCSDresultsandallCASSCFmethodsisrathergoodth oughhigherthanthe STEOM-CCSDresults. ExcitedstatepropertiesaresummarizedinTable 2-2 .ThegroundstateEDMfor phenolis1.32DandthegroundstateEDMforphenol-NH 3 is3.97Dforcorrelated methods.Thegroundstater 2forphenolis705.9andthegroundstater 2forphenol-NH 3 is1469.4.Thedifferencebetweenthesizeofthegroundstat eandtheexcitedstate, r 2isreportedforEOM-CCSDandSTEOM-CCSDresults.Large r 2valuesrelative tothegroundstateindicateRydbergstateswhilesmall r 2valuessignifythatthe excitedstatesarevalence-typetransitions.The r 2valuesofCTstatestendtofall in-between,butinthiscasetheCTstatealsohappenstobeaR ydbergstate.TheR 131

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Figure2-5.VerticalexcitationspectrausingtheCIS,TD-D FT/B3LYP,EOM-CCSD,and STEOM-CCSDmethodslistedinTable 2-2 .Theexperimentalcurveis adaptedfromKimuraandNagakura[ 69 ]whichshows 1and 2statesof phenolinheptanesolventttedwithaGaussianlineshape. 32

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maybecharacterizedasachargetransferstatebetweentheb enzylandthehydroxide groupsbasedonthe r 2valueandthemolecularorbitalsinvolved.Itisrelatedtot heR x 1CTstatebetweenphenolandammonia.Notethelargechangein EDMforthisstateas theelectrondensitymovesfromthe systemtoa orbital. Thethirdexcitedstate(R 2)offreephenolalsohasalargeincreaseinEDMdue toanappreciablemovementofelectrondensitybetweenthe systemandthediffuse orbital.However,theaddedammoniamoleculeisnotinvolve dinthisstate.Forthe phenol-NH 3 complextheEDMsubstantiallyincreasesfortheR x 1stateduetoelectron densitymovementbetweenthesolventammoniaandthephenol .Therefore,whilea largedifferenceinEDMcanbeanindicatorofCTbetweensolv entandsolute(R x 1)itcan alsobeanindicatorofCTtoadiffuseorbital(R 1,R 2,R x 2). 1, x 1, 2and x 2statesshowamodestincreaseinEDM.TheincreaseinEDM followsfromchargere-distributionontheringcausingcha rgebuild-upatoneormore atomiccenters.Theseobservationsshowthatthechangesin EDMareverysusceptible toelectrostaticsandCTisnotalwaysnecessarytoinvokeif alargechangeinEDMis found. ThechangeinEDMisregardedasthemainreasonfortheexcita tionenergyshifts ofmoleculesinsolution.ThechangeintheEDMbetweenthefreemoleculeandits ammoniacomplexis:( EDM k ) = ( EDM k ) c( EDM k ) f(2–1) foreachk thexcitedstate. NotethattheEDMforbothRydbergstatesisnegativesignifyingthatther elative dipolemomentofthecomplextightenswithrespecttotherel ativeEDMofthefree molecule.FortheR 1state,whichisrelatedtotheR x 1CTstate,theEDMis-4.22D whilefortheR 2state,whichisrelatedtotheR x 2state,theEDMisonly-1.99D.This resultfollowsfromR 2notbeingasaffectedbytheadditionofsolventandretainin gits 33

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orbitalcharacter(refertoFigure 2-4 )whilethecharacteroftheR 1statechangesfrom involvingadiffuse orbitalonthephenoltoa orbitalontheammonia.TheEDM valuesforthetransitionsarepositivebutsmallinmagnitudecomparedto thoseofthe Rydbergstatessignifyingthattheadditionofonesolventm oleculecausesmoderate changesinthechargedistributionoftheseexcitedstates. Alloftheexcitedstates showanenergyred-shiftassociatedwiththeadditionofamm onia,however,thesignofEDMmaybepositiveornegative. ThepropertiescalculatedwiththeEOM-CCSDmethodarecons istentwiththose calculatedwiththeSTEOM-CCSDmethod.TheEDMvaluesforSTEOM-CCSDtend tobeslightlylargerinmagnitudethanthoseforEOM-CCSD.T heoscillatorstrengthalso tendstobegreaterthantheEOM-CCSDcalculation.Theresul tsinFigure 2-5 slightly favortheSTEOM-CCSDmethodbuttheoscillatorstrength, r 2,andEDMvaluesare notradicallydifferentbetweenthetwo.TheSTEOM-CCSDcal culationscalesbetter withsystemsizethanEOM-CCSDandoffersnotonlyapossible accuracyboostbuta computationaladvantageaswell.2.4.2 Abinitio spectraof -naphtholand -naphthol Thegeometriesofthe and isomersoffreenaphtholandtheammonia-naphthol complexareoptimizedattheMBPT(2)leveloftheoryusingth eparallelversionofACES. Only24to48processorsareuseddependingonresourceavail abilityandnearto linearscalingwasobservedacrossdifferentnumberofproc essorsincaseswherethe samecalculationhadtoberepeatedondifferentnumbersofc ores.Thesinglepoint energyiscalculatedattheCCSDandCCSD(T)levelsoftheory .Thedesignationsused fortheisomersare Nfor -naphthol, NCfor -naphthol-ammoniacomplex, Nfor -naphthol,and NCfor -naphthol-ammoniacomplex.Thegroundstateenergyvalues fornaphtholandtheammonia-naphtholcomplexesarelisted inTable 2-3 Theimportanceofusingacorrelatedmethodforthegroundst ateisillustratedin Table 2-3 .TheHartree-Fockenergyshowsthe isomertobelowerinenergythan 34

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Table2-3.ThegeometriesareoptimizedatMBPT(2)/6-31++G **leveloftheory.All energyvaluesforpost-HartreeFockmethodsarereportedas correlation energies.Theenergyisreportedinatomicunits.The Ebetweenthe Nand Nandthe NCand NCisreportedinkcal/molunits. Method N NC N NC E HF-458.23920-514.45124-458.23955-514.45131 E0.220.0400E MBPT (2)-1.51927-1.71753-1.51860-1.71706 E000.200.25E CCSD-1.56343-1.77486-1.56292-1.77453 E000.100.16E CCSD ( T )-1.63505-1.85147-1.63445-1.85105 E000.160.22 H24 N20 H19 O18 H16 H15 H11 H17 H12 H13 H14 C9 C8 C7 C6 C5 C4 C3 C2 C1 C10 H21 H22 A NC N20 H19 O18 H16 H15 H11 H17 H12 H13 H14 C9 C8 C7 C6 C5 C4 C3 C2 C1 C10 H21 H22 H23 B NC Figure2-6.TheatomiclabelscorrespondingtoTable 2-4 the isomerbutwiththeadditionofcorrelation,therelativeen ergiesarereversed. Quantitatively,therelativeenergiesoftheCCSD(T)metho dareclosertothoseofthe MBPT(2)methodmakingitourchoiceforgeometryoptimizati on. Theoptimizedgeometricparametersforthenaphtholmolecu lesandtheirammonia complexesarelistedinTable 2-4 andthelabelsareshowninFigure 2-6 35

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Table2-4.Optimizedgeometryparametersforfreeandcompl exed -naphtholand -naphtholwithMBPT(2)/6-31++G**.ThedistancesareinAng stromandthe anglesareindegrees. Coordinate N NC N NC C 1 C 21.42051.42741.42121.4211C 2 C 31.41971.41841.41891.4188C 3 C 41.38171.38261.38201.3823C 4 C 51.41471.41491.41491.4150C 5 C 61.38241.38191.38231.3825C 6 C 71.41851.42011.41981.4204C 7 C 81.42351.42021.41971.4192C 8 C 91.38181.38611.38111.3854C 9 C 101.41451.41391.41551.4186C 10 C 11.38001.38611.37821.3782C 10 H 111.08301.08231.08251.0828C 3 H 121.08451.08171.08441.0846C 4 H 131.08311.08341.08311.0832C 5 H 141.08321.08331.08321.0834C 6 H 151.08161.08481.08471.0851C 8 H 161.08481.08391.08591.0843C 9 H 171.08371.08361.08431.0846C 1 O 181.37801.36441.37661.3642O 12 H 130.96720.96670.96710.9864H 13 N 14-1.8641-1.8678N 14 H 15-1.0139-1.0141N 14 H 16-1.0142-1.0142N 14 H 17-1.0142-1.0142 \C 1 C 2 C 3121.9121.5122.2122.3 \C 2 C 3 C 4121.0120.3120.7120.8 \C 3 C 4 C 5120.2120.5120.2120.1 \C 4 C 5 C 6120.6120.1120.4120.3 \C 5 C 6 C 7120.1121.1120.8121.1 \C 6 C 7 C 8122.0121.9121.9121.9 \C 7 C 8 C 9121.1120.0120.3120.6 \C 9 C 10 C 1120.7120.4119.9120.5 \H 11 C 10 C 1120.2119.6121.5121.3 \H 12 C 3 C 4120.3121.0120.4120.4 \H 13 C 4 C 3120.0119.8120.0120.1 \H 14 C 5 C 6119.7120.0119.9119.9 \H 15 C 6 C 5120.8120.2120.2120.1 \H 16 C 8 C 9120.0120.7120.5120.2 \H 17 C 9 C 8120.6120.0119.8119.9 \O 18 C 1 C 2115.9116.4123.5123.9 \H 19 O 18 C 1108.9110.8109.2110.9 \N 20 H 19 O 18-169.5-169.9 \H 21 N 20 H 19-118.3-118.0 \H 22 N 20 H 21-107.1-107.1 \H 23 N 20 H 21-107.1-107.1 ]H 22 N 20 H 21 H 19--122.6--122.7 ]H 23 N 20 H 21 H 22--114.7--114.7 36

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Thepresenceofthesolventammoniamoleculehastheexpecte deffectonthe geometryof -naphthol.Thecarbon-carbonbondsarelengthenedbyanave rageof 0.009inthecomplexbutthegreatestdeformationisexperi encedbytheCObond (contractedby0.0124)andtheOHbond(expandedby0.0193 ).Thiskindof geometrydeformationwouldcontributetothe586cm -1 red-shiftobservedbyFleisher etal.. aswellastheobservedincreaseintheEDMforgroundandexci tedstates. Thischangeingeometryisnotincludedinthedipolemomentv ectoradditionanalysis sincethefree -naphtholusedtherewasatitsequilibriumgeometry.Theam monia solventhasaslightlydifferenteffectonthe -naphtholisomer.TheCObondcontracts by0.0136buttheOHbondcontractsby0.0005,thuseventho ughtheCObond strengthenswiththeadditionofsolvent,theOHbonddoesno tweaken.Theoptimized geometryparametersareconsistentwiththeprevioustheor eticalcalculations[ 56 104 150 ]withtheexceptionoftheresultsfromDFTmethods.The NCintermolecular Hydrogenbondwascomputedtobe1.789withthePW91/6-31G( d,p)method (Henseler etal.. )and1.837withtheB3LYP/6-311++G(d,p)method(Tanner et al.. ).TheMBPT(2)calculationofHenseler etal.. with6-31G(d,p)basissetyielded aHydrogenbondof1.919.Thisindicatesthattheaccuracyo ftheintramolecular bonddistanceisasaffectedbybasissetasitisbythechoice of abinitio method.The Hydrogenbondof NCwascalculatedtobe1.905byMatsumoto etal.. withthe HF/6-31Gmethod. Table 2-5 liststhe abinitio excitationenergyvaluesaswellasseveralpropertiesof thethreelowestexcitedstates.Thedominantorbitalsfore achexcitationcorrespondto theorbitalsshowninFigures 2-7 and 2-8 .ACTstateforthenaphtholcomplexwould involveatransitiondominatedbyeitheranorbitalontheam moniagoingtoanorbitalon thenaphtholoranorbitalonthenaphtholgoingtoanorbital ontheammonia,similar tothesituationforphenol.TheR 1andR x 1stateshowCTbehavioraccordingtothe 37

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Figure2-7.Hatree-Fockorbitalsof -naphtholcorrespondingtotheexcitationslistedin Table 2-5 molecularorbitalanalysis.AllunitsanddesignationsinT able 2-5 arethesameasthose inTable 2-2 ThediscrepancybetweentheEOMandtheSTEOMexcitationene rgiesisworse forthenaphtholmoleculesthanforthephenol.The transitionsonthenaphthol moleculeshavemoredoubleexcitationcharacterthantheon esonthephenolduetothe systembeingmoreextensive.EOM-CCSDperformsmuchbetter forsingleexcitations thanforthosewithappreciabledoubleexcitationcharacte r.Triplesexcitationsare requiredtoimprovethelatter[ 120 ].TheagreementinenergyistheworstfortheR 1stateof -naphthol.Inthiscase,itismostlikelythattheSTEOM-CCS Dvalueis about0.1eVtoolow.Thisistheonlystatewherethe%singles(aprojectionofthe 38

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Figure2-8.Hatree-Fockorbitalsof -naphtholcorrespondingtotheexcitationslistedin Table 2-5 STEOM-CCSDwavefunctiononaspaceofsingly-exciteddeter minants)isaslowas 71%whereasforalloftheotherstatesitis88%andabove.TheSTEOM-CCSDEDM valueforR 1isalsounderestimated:theEDMforR 1is10.95DwithEOM-CCSD whichexceedstheerrorrangeshowninTable 2-2 forphenol.Generally,whenthe%singlesvaluedipsbelow85%,atriplescorrectionisnecessarytorescuetheenergyand propertiesvaluesforthatstate. ThechangeinEDMoftherstexcitedstateof -naphtholisdifferentthanfromthe CIScalculationsofFleisher etal.. .TheEDMvaluesfromFleisher etal. are0.16D forthefree -naphtholand1.05Dfortheammoniacomplex.Asshowninthis work, theCISmethodlacksthenecessaryaccuracytocomputethesm alldifferencesin 39

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Table2-5.TheexcitationenergyisreportedineV.Theoscil latorstrength(f)isunit-less. ThevaluesforEDMarereportedinDebye.Theexcitedstatelabelsand orbitalscorrespondtoFigure 2-7 andFigure 2-8 StateEOM-CCSDSTEOM-CCSD EE(eV)fEE(eV)f EDM ( D ) r 2 N 14.330.0093.870.0060.490.6 24.890.0934.660.1243.452.4R 15.220.0004.970.0008.3454.0 NC x 14.300.0123.840.0110.563.3 x 24.760.0844.500.1354.017.6R x 15.040.0004.950.0007.5167.7 N 14.290.0153.810.0150.34-0.4 25.100.0524.900.0750.691.2R 15.330.0005.230.00011.5441.9 NC x 14.220.0263.760.0210.85-0.2 x 25.050.0484.820.0722.033.3R x 15.130.0005.000.0008.0050.5 properties.UsingtheSTEOM-CCSD/DZP++leveloftheory,we computetheEDM of0.34Dforthefree -naphtholand0.85Dfortheammoniacomplex;amuch lessdramaticdifferencewithEDMofonly0.51Dwhichismoreconsistentwith Suppan[ 149 ].OnehalfofaDebyedifferenceisactuallysmallwhencompa redtoaEDMvalueofatrueCTstatesuchasR 1. 2.4.3Comparisontoexperimentalresults Therst(andinsomecasesthesecond)excitedstatesofphen ol, -naphthol, -naphtholandtheirammoniacomplexeshavebeendocumented experimentallywith avarietyofmethodsinthegasphase,Argonmatrix,andinsol ution.The 1statehas beenprobedbybothabsorptionandemissionspectroscopict echniquesandthe0-0 vibronicbandreported.Thecollatedexperimentalresults arepresentedinTable 2-6 TheagreementbetweentheSTEOMvaluesandtheexperimental valuesfortherst excitedstateisexcellent,despitetheexclusionoftheadi abaticcorrectionfromvertical excitationtothe0-0vibronicexcitation.Wealsohavegood agreementwiththeresults fromnon-polarsolventsaswellastheArgonmatrixexperime nts.Forthesecondexcited 40

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Table2-6.Theexcitationenergiesfromexperimentsarepre sentedforrstandsecond excitedstatesofphenol(Ph),phenol-NH 3 (PhC), -naphthol( N), -naphthol ( N), -naphthol-NH 3 ( NC),and -naphthol-NH 3 ( NC).TheunitsareeV andthesourcesofthevaluesare:a:[ 65 68 119 154 ];b:[ 127 ]; c:[ 65 68 143 ];d:[ 24 25 58 ];e:[ 67 104 ];f:[ 117 ](heptane); g:[ 24 25 58 63 72 ];h:[ 144 ](hexanes);i:[ 36 ];j:[ 19 ](Armatrix);k:[ 44 129 ]; l:[ 69 ];m:thiswork(verticalexcitation)STEOM-CCSD/DZP++;n: thiswork (verticalexcitation)EOM-CCSD/DZP++ StatePhPhC N NC N NC 14.51a4.20b3.90d3.87g3.75h3.61h4.43c3.87e3.83e3.72i3.85f3.79i3.76j3.81j3.76k3.78f4.41m4.34m3.87m3.85m3.81m3.76m4.90n4.83n4.33n4.30n4.29n4.22n 25.77l4.28f4.35f6.06m4.66m4.82m6.23n4.89n5.10n statetheagreementislessclearsinceweonlyhaveanexperi mentalresultinsolution. Therelativeenergyforthesecondexcitedstatecompareswe ll:N 2
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forreasonsexplainedintheprevioussection.Sincetherei snoactivespacechoicein theEOMscheme,theenergydifferencesareincloseragreeme ntwithexperiment.The agreementinenergiesisrathergoodconsideringthat100cm -1 isequivalentto0.01eV whichexceedstheaccuracyofanyofthesecalculations.How ever,wecanhopethatthe errorwilllargelycancelforrelativeenergyvalues.Thisk indoferrorcancellationallows obtainingreasonableagreementbetweenthecalculatedred -shiftsandtheexperimental red-shifts.TheTD-DFT/B3LYPvaluesshowlittleconsisten cy:theEofthe 1stateof phenolis-879cm -1 Thered-shiftduetotheammoniasolventoftherstexciteds tate( 1)maybe comparedwithexperimentalvalues.TheSTEOMandEOMvalues forphenoland -naphtholareunderestimatedbyabout100cm -1 ,however,thereisalsoalarge variationinthesevaluesamongthevariousexperimentalre sults,especiallyfor the -naphthol,perhapsduetosomedifcultiesintheassignmen tofthe cis/trans isomers[ 36 104 ].Thestandarddeviationofthered-shiftismuchlessforth e -naphthol isomer.Despitetheseissues,therelativeenergiesagreev erywell:thered-shiftof 1of -naphtholisabouttwicethemagnitudeofthered-shiftof 1of -naphtholforthe theoreticalcorrelatedmethodsaswellasforexperiment.T hevaluesfromtheB3LYP methodshownocorrelationtoanyexperimentalorcorrelate dmethod.Severallarge basissetcalculationsofexcitedstateenergieswereperfo rmedforphenolbuttheywere notsignicantlyclosertotheexperimentalvaluesthanthe onesdonewithDZP++basis. Therefore,wedeemDZP++adequateindescribingtheseveral lowestexcitedstates. 2.5Conclusion Recenthigh-resolutionspectroscopystudiesof -naphthol, -napthol,andphenol showasolvatochromiceffectintherstexcitedstatewithj ustonesolventmolecule present.Thispresentsanopportunitytocompareexperimen talsolventeffectswith ab inito theorywithoutmakingcrudesolventmodelapproximations. Wecanalsobeginto applytheterm“solvatochromic”,whichisgenerallyreserv edforthecondensedphase, 42

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Table2-7.Theexcitationenergydifferencesbetweentheph enol-ammoniacomplexand freephenolandtheexcitationenergydifferencesbetweent he and naphthol-ammoniacomplexandfreenaphtholarereportedin wavenumber units.Theexperimentalvaluessourcesare:a:[ 65 ],b:[ 68 ];c:[ 143 ];d:[ 58 ]; e:[ 24 25 ];f:[ 44 67 129 ];g:[ 36 ];h:[ 19 ](Armatrix). StateEOM-CCSDSTEOM-CCSDEXP EE EE EE PhC 1-581-564-635a,-642b,-650c R 1-1403-1588R 2-2719-2882 2-1528-1778 NC 1-243-177-240d,-236e 2-1096-1286R 1-1467-205 NC 1-505-432-586f,-606g,-409h 2-453-637R 1-1623-1811 tothesegas-phaseexperimentsand abinitio calculationssincethereareobvious solventeffectspresent. Basedonanorbitalanalysisofthegroundandexcitedstates ,itisdeterminedthat onlyamarginalchargetransferoccursbetweenthesolventa ndthesolutemolecules forthersttwo states.TheampliedchangeinEDMinthepresenceofone ammoniasolventmoleculepointstosomedifferencesinchar gedistributiononthe naphthol(orphenol)fragmentofthecomplexbutnotduetoch argetransfer. WealsoshowthattheCISmethod,whichisoftenusedfor abinitio spectroscopy oflargermolecules,doesnotpossessufcientaccuracybas edon(1)itscomparison withhigherleveltheoryandexperimentalvaluesand(2)the incorrectdescriptionof theHatree-Fockgroundstateof and isomersofnaphthol.Ifoneiscomputationally restrictedtoasingleparticletheoryevenTD-DFTisabette rchoiceprovidedthat noCTstatesareconsideredandwiththecaveatthatoscillat orstrengthsmaybe questionable.TheSTEOM-CCSDmethodoffersacomputationa ladvantagesince itscalesas 2 n 5asopposedtoEOM-CCSDwhichscalesas n 6wherenisthe 43

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numberoforbitals.ThecalculationoftheCCSDgroundstate wavefunction( n 6)is requiredforbothmethods.STEOM-CCSDalsooffersadvantag esforexcitedstates withsignicantmixinganddouble-excitationcharactersu chasthe states inthisstudy.However,forcomputingsmalldifferences(on thescaleoflessthan0.2 eV)inenergybetween different molecules,theEOM-CCSDmethodoffersapotential advantageoverSTEOM-CCSDbecauseacompleteorbitalspace isalwaysincludedin theEOMframeworkwhileSTEOMdependsonaselectionoforbit als. 44

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CHAPTER3 ABSORPTIONCROSSSECTION 3.1Background Thephotodissociationratesofvolatileorganicandinorga niccompoundsare criticallyimportantinmodelingthecompositionoftheatm osphere,inaddressingglobal warming,ozonedepletion,andotherphenomena.Theabsorpt ioncrosssectioninthe spectralrangeofsolaruxisneededtocalculatethephotod issociationrateconstantJ =Z Fd (3–1) where istheincidentwavelength, istheabsorptioncrosssection, isthequantum yieldofphotodissociationandFisthesolaractinicux. Thephotodissociationrateconstantanditsdependenceont emperaturecan bemeasuredprovidedapuresampleisobtainedandtheabsorp tioncrosssection atvarioustemperaturesisknown[ 100 105 ].Thepuresampleconditionbecomes increasinglydifculttosatisfyasthesize,complexityan dstabilityofthecompoundin questionimpedeattemptstosynthesizeit.Currently,thea bsorptioncrosssectionsthat areusedtodeterminethephotodissociationratesofcomple xorunstablemolecules areunknownandarecrudelyestimated.Anexampleoftheprob lemsarisingfrompoor estimatesarethephotodissociationratevaluesfororgani cperoxidesthataredirectly derivedfromtheabsorptioncrosssectionsofhydrogenpero xideandmethylperoxide simplyduetotheirexperimentalavailability,bypassingt hedifcultiesinobtainingthe experimentalcrosssectionsoftheactualmolecules[ 66 ].Usingestimatedvaluesas opposedtothetruevaluescanleadtoseriousinaccuraciesi nthesteadystatemodels oftheatmosphere. Normally,whenanabsorptioncrosssectioniscalculatedwi th abinito methodsa vibronicmodelcanbeusedtoincludesomemolecularmotion. Inordertocompute aphotodissociationabsorptioncrosssectionforrelative lylargespeciesseveral 45

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complicationsofthevibronicmodelneedberesolved.Noneo fthecurrentlyavailable softwarecanproperlyhandlethedissociativeexcitedstat esurface,andfewcan correctlyhandleatorsionalpotentialsinceitdoesnotnat urallylenditselftoaquadratic orquarticexpansionbutrathershouldbeexpandedwithaset oftrigfunctions[ 43 75 ]. EvenifthismodelismadetoworkwiththetypesofPESsinvolv edinphotodissociation, thereisstilltheaspectofndingrootsofaverylargematri xwhichmaynotbesparse whentherearemanystrongvibroniccouplingspresent. Themostwidelyusedprogramsthathavethecapabilitytosim ulateanabsorption spectrumareVIBRON[ 121 ]andHOTFCHT[ 14 ].Alloftheseprogramsrequirea calculationoftheharmonicnormalmodesoftheabsorbingst atewhichworkswell enoughforexcitedstatepotentialenergysurfaceswhichha veastationarypoint,but donotperformwellfordissociativepotentialenergysurfa ces.TheLEVEL[ 82 ]program worksforboundandquasi-boundpotentialsandhasbeensucc essfullyappliedtoa varietyofdiatomicmolecules. 3.2Methods 3.2.1Approachtotheproblem Inordertocalculatetheabsorptioncrosssectionforlarge moleculesseveral simplicationsneedtobemade.Therecurringthemeisthatw earetryingtoremove asmanynon-vitalvibrationaldegreesoffreedomaspossibl e.Theseassumptionsare explainedbelowandwillbeillustratedthroughouttheNaOH example. 1. Considerthedissociativecoordinateastheprimarycoordi nate. Thisistheobvious choicesinceweareinitiallyinterestedinthosecrosssect ionswhichleadto dissociation.Thismethodcanbeextendedtouseanydegreeo ffreedomasthe primarycoordinate. 2. Computetheabsorptioncrosssectionoftheprimarymode. Thiswillbethe zeroth-ordersolution.Thediscretevariablerepresentat ionmethod(DVR)isusedto computetheFrank-Condonoverlapintegralsforadissociat ivemode. 3. Consideronlythosevibrationswhicharesignicantlyther mallypopulated. In practice,thiswouldinvolveasubsetofvibrationsofenerg yequalorlowerthan 46

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theenergyofthestretchalongwhichdissociationtakespla ce.By equal wemean withinsometolerancewhichisoptimalforagiventemperatu re.Forexample,ifone isinterestedingoingtoveryhightemperatures,modeshigh erinenergythanthe primarymodeshouldbeconsidered. 4. Determinetheimpacteachsecondarymodehasontheexcitati onenergiesof interestaswellasthecorrespondingdipoletransitionmom ents. Iftheimpactis lessthansometolerance,removethatnormalmodefromthesu bset.Thiswill nowdenethesubsetofsecondarymodes.Anydegreeoffreedo mwhichis notexplicitlyapartofthesecondarymodes'subsetwillber elaxedinallpartial geometryoptimizationcalculationssothatanysmalleffec tthesemodeshavecan bepartiallyreectedinthenalanswer. 5. Addtheeffectsofotherdegreesoffreedombyperturbingthe primarymode groundstateandexcitedstatepotentials. Thiscanbedonebyextrapolatingor ttingthecurvesusedtodeterminetheimpactsinstepIV. 6. Temperatureeffectscanbeintroducedatthisstage. Thisstepinvolvesredistributing energyquantaamongthevibrationallevelsoftheprimarymo deandthevarious secondarymodes. 3.2.2Discretevariablerepresentation Thediscretevariablerepresentation(DVR)method[ 90 ]hasbeenusedtosolve thevibrationalHamiltonianforavarietyofsmallchemical compoundsprovideda potentialenergysurfaceisknownaswellasexpandedtowork uptoN=16vibrational degreesoffreedom[ 132 133 164 ].Itcanbeusedtondvibrationalwavefunctions onadissociativesurface[ 29 110 ].DVRworksbyanumericaldiscretizationofthe kineticenergyoperatorandthepotentialintoNsegments.Thedetailedderivationofthe DVRusingtheFouriergridHamiltonianmethodcanbefoundel sewhere[ 29 103 ].The discretizedformofthekineticenergyoperatorisshownbel owwheremisthereduced massand xisthesizeofthegridonwhichthepotentialsareplaced.T ii0=~2 2 m x 2 (1) ( ii0) 8><>: 2 3fori = i02 ( ii0) 2fori6= i0 (3–2) 47

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TheHamiltonianthatisdiagonalizedisshownbelowwhereVisthepotentialfunction usedtotthe abintio datapoints.TheHamiltonianbelowininatomicunits.H ii0= T ii0+ V ii0 ii0= (1) ( ii0) 2 dx 2 8><>: 2 3fori = i02 ( ii0) 2fori6= i0 9>=>;+ V ii0 ii0 (3–3) Theresultsaresensitivetotheresolutionofthegrid.Sinc ethisisaveryfastcalculation, itiseasytovarythenumberofgridpointsuntiltheeigenval uesintheenergyrangeof interestconverge.Inthiswork,501gridpointsareenought oreachconvergence. 3.2.3Discreteabsorptioncrosssection OnecanrewriteEquation 3–1 toincludethetemperaturedependenceofthe photodissociationrateconstant:J =Z T Fd (3–4) wherethetemperaturedependenceisaddedviatheabsorptio ncrosssectionterm.The absorptioncrosssectioncanbewrittenas: (, T ) = KXkfh kj j 0iMXm =0 N ( T )Xn =0h m kj n 0ig (3–5) where istheelectronictransitionmomentobtainedfromtheelect ronicwavefunctions,andarethevibrationalwavefunctions.ThesummationsareoverK-electronic excitedstates,M-vibrationalstatesontheexcitedPESandN-vibrationalstatesonthe groundPESwhereN ( T )isthemaximumoccupiedvibrationallevelsattemperature,T.Thetemperaturedependenceoftheabsorptioncrosssectio ncomesfromvarying thepopulationofthegroundstatevibrationalenergylevel s.EOM-CCSDwillbeusedto computetheelectronictransitionmomentandaDVRcodedeve lopedbytheauthorwill beusedtocomputetheFrankCondonintegrals. Inordertofacilitatecomparisonbetweentheexperimental andthetheoretical absorptioncrosssectionsitisbesttoexpressallintensit iesasunit-lessoscillator 48

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strengthsasopposedtotransitiondipolemoments.Therela tionshipbetweenthetwo quantitiesis:f 0 k =f82 m e cg n 3 he 2g !0 k0 k(3–6) !0 kistheexcitationenergy, 0 kisthedipolestrength,cisthespeedoflight,eisthe elementarycharge,m eisthemassofanelectron,histhePlanckconstant,andg nisthe electronicdegeneracyoftheexcitedstate. Theexperimentaldatamayalsobeexpressedastheoscillato rstrength[ 22 ] providedthatthemolarextinctioncoefcientisknown:f 0 k =f2303 m e c 2 N A e 2gFZ !2!1d! (3–7) Here,N AisAvogadro'snumber,Fisthecorrectionfortherefractiveindex,whichcan besetto1sincethisappliestoadiffusegas,and isthemolarextinctioncoefcientin theunitsofL=molcm.ThequantitiesobtainedfromEquation 3–6 andEquation 3–7 theexperimentalandtheoreticalverticalexcitationinte nsitiesmaybecomparedandthe agreementisknowntobegoodbasedonexamplescurrentlyfou ndintheliterature[ 17 45 ]. 3.2.4CombineDVRandthediscreteabsorptioncrosssection Caremustbetakentoproperlycombinetheelectronicstruct ureresults(oscillator strength)withtheDVRresults(FrankCondon)suchthatboth areexpressedasa functionof andlieonthesamegrid.Thereareseveralwaystoachievethi sandone suchwayisdescribedhere. SolvingtheHamiltonianaswritteninEquation 3–3 willproducethefollowingsetof FrankCondonintegrals:FC im =f Xjmjjig2(3–8) where mjistheeigenvectorcorrespondingtothem thenergylevelofthegroundstate potentialenergysurfaceand jiistheeigenvectorcorrespondingtothei thenergylevel 49

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oftheexcitedstatepotentialenergysurface.Theseintegr alsarenormalizedsuchthat Pi FC im = 1.Thedifferenceofeigenvaluesforthegroundandexcitedst ateareusedto buildthe !ivector:E iE k =!i. Theoscillatorstrengthsarecalculatedasafunctionofgeo metryforeachelectronic transition.However,onlythevalueattheequilibriumgeom etrycorrespondstothe experimentalvaluedeterminedusingEquation 3–7 ,whichisanintegralofthecross section.Whatisneededisawaytomaptheoscillatorstrengt hvaluetotheareaofthe crosssection,preferablyincgsunits.First,rearrangeEq uation 3–7 toreectthecgs units:f 0 k =fm e c 2 e 2gFZ !2!1d! (3–9) where istheabsorptioncrosssectionincm 2=molecule.Now,rearrangetosolve for discretely:~i =f e 2 g n m e c 2gF i !i +1 !i(3–10) Notethatthis~ isadiscreteelectronicpartofthetotal .The inthedenominator comesfromtheeigenvaluesoftheDVRHamiltonian.Theoscil latorstrengthisalso discreteandobtainedfrom abinitio calculationswhichiswhyg nappearsinthis equation.Atthispoint,F iistheoscillatorstrengthwhichcorrespondstoeachm!itransitionandisestimatedbyaweightedaverageoscillato rstrength, f i: f i =Pj f ij mjjij Pjj mjjij (3–11) CombiningEquations 3–10 and 3–11 allowstheelectronicpartof tobeplaced onthesamegridastheFranck-Condonpartof suchthatthelargestcontribution toF iarecomingfromtheportionsoftheoscillatorstrengthsurf acewiththegreatest Franck-Condonoverlap.Thetotalabsorptioncrosssection foreachelectronictransition 50

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whichcanbecomparedwithexperimentisthenobtainedbycom biningtheelectronic andvibrationalparts: 0 k =Xi ~i Xm c T m FC im(3–12) Thesubscriptmwillbegreaterthanonewhenmorethanonegroundstatevibra tional energylevelispopulated.Thepopulationofthem-statesar eweightedbyasetof temperature-dependentconstantsc m,whichcorrespondtotheBoltzmanndistributionat agiventemperature.InthecaseofT = 0 K,mwillalwaysbesetto1correspondingto therstvibrationallevel. Thenalcrosssectionsforeachk thexcitedstatearetobeconvolutedtoobtaina theoreticalspectruminthedesiredspectralrange. Oneconsistencycheckfortheprocedureoutlinedaboveisto useEquations 3–7 or 3–9 tondtheoscillatorstrengthoncethecalculatedabsorpti oncrosssectionis obtained.Ifeverythingisdonerightinthecode,thetstot hePESareadequate,and thegridisneenough,thef DVRshouldbeclosetothecalculatedfforeachtransition. Furthermore,improvingupontstothepotentials(byusing morepointsforexample) aswellasincreasingtheresolutionofthegridshouldconve rgethef DVRtothecorrect value. 51

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CHAPTER4 ABSORPTIONCROSSSECTIONEXAMPLES 4.1SodiumHydroxide Weproposeamethodforcomputingabsorptioncrosssections fordissociative surfaceswhicharegearedtowarddescribingalargermolecu lewithmorethan10 vibrationaldegreesoffreedom.Asanillustrationoftheme thodologywechooseasmall molecule,sodiumhydroxide.Photodissociationofgaseous NaOHaswellasother sodiumoxidecompoundsisanimportantatmosphericprocess inthemesosphere[ 128 ], andtheabsorptioncrosssectionsleadingtophotodissocia tionhavebeenobtained experimentallyusingameabsorption[ 2 ]andlaserspectroscopy[ 139 ].Thepertinent experimentalresultsarefromthelaserspectroscopystudy of SelfandPlane 4.1.1Electronicstructurecalculations Thestartingpointforcalculatinganabsorptionspectrumi stobeginwithsingle pointverticalexcitationenergies.Fromtheenergyrangeo ftheexperimentalspectrum inFigure 4-1 ,weseethatthreelowenergyexcitedstatesarerequiredtoc reatea theoreticalspectruminthenearUVspectralrange.Thesest ates,theirsymmetryand characterarelistedinTable 4-1 .Theerrorbarsfromtheexperimentaldatashowagreat dealofuncertaintyintheintensityespeciallyinthepeaka t220nm.Inthisexamplethe agreementbetweenthecalculatedandthetheoreticaloscil latorstrengthisnotasgood asithasbeenforothermoleculesfoundintheliterature[ 17 45 130 ].Ahighresolution absorbtioncrosssectionofNaOHwouldbedesireabletoobta inforcomparison.Asit presentlystands,thecalculatedoscillatorstrengthisab outtwotimesgreaterthanthe experimentalvalue.Table4-1.Characteristicsofthelowenergyexcitedstates ofNaOH. StateSymmetryCharacterTypeElectronicEOM-CCSD/f fDegeneracyPOL(nm)calc.exp. A1 (0)!1 (1)Valence 2344.50.0540.017 B1 (0)!1 (2)Rydberg 2239.20.0920.060 C1 (0)!1 (1)Rydberg 1227.30.026 52

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nn nr r r r rnr nrn nnr nnrr Figure4-1.ThePOLbasissetdoesareasonablejobofdescrib ingtheRydbergexcited statesinthe220nmregionbutitunderestimatestheenergyg apofthe valencestate.TheresultsfromEOMandSTEOMcalculationsw ithWMR basisareingoodagreementwitheachotheraswellaswiththe experiment. Thedipoletransitionmomentsremainlargelyunaffectedby thechoiceof basisset.Verticalexcitationintensitiesareapproximat ebutreectthe theoreticalrelativeintensities. ThegeometryisoptimizedattheCCSDleveloftheoryusingth ePOLbasisset [ 136 ].TheR NaO=1.9493andR OH=0.9589.ThebestavailableexperimentalNaOH geometry[ 79 ]isR NaO=1.9500.Thisgeometryisalsoingoodagreementwiththe geometrieslistedintheworkof LeeandWright whoemployavarietyofhigh-end methodsandbasissets[ 83 ]whichsuggeststhatCCSD/POLisadequatetoproceedas farasgeometryisconcerned.Keepinginmindthatthemethod outlinedbelowisgeared forlargermolecules,thebestoptimizedgeometrywillofte nbeaDFTgeometrydone withamodestbasisset. TheexcitationenergiesarecalculatedusingtheEOM-CCSDm ethodandPOL basissethasproventobeagoodchoicefordescribingexcita tionenergiesanddipole 53

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Table4-2.Near-UVEOM-CCSDexcitationenergiesforNaOH. StatePOLaug-cc-pVDZaug-cc-pVTZWMRMADExp. eVnmeVnmeVnmeVnmeVnmnm A3.60344.503.65340.103.81325.703.84322.700.109.0031 3 B5.18239.205.18239.305.34232.105.42228.300.104.5023 0 C5.45227.305.45227.505.61220.905.70217.900.104.0022 5 Table4-3.Near-UVSTEOMexcitationenergiesforNaOH. StatePOLaug-cc-pVDZaug-cc-pVTZWMRMADExp. eVnmeVnmeVnmeVnmeVnmnm A3.53351.103.58346.503.76329.403.78327.900.1110.103 13 B5.15240.605.16240.205.30233.705.41229.300.104.4023 0 C5.26235.605.44227.705.59221.705.66218.900.145.7022 5 momentstrengths.SinceNaOHisasmallmolecule,wearealso abletocalculatethe spectrumwiththeaug-cc-pVXZ(X=D,T)[ 37 161 ]basissetsof DunningJr. andwith theWMR[ 158 ]basissetofWidmark etal. Thismaynotbepossibletodoforthelarge moleculesthatareourobjectivesowewillalwayspresentth eEOM-CCSD/POLresult asanexpectedelectronicstructurelevelofaccuracyalong withanyenergycorrections madetoitbasedontheresultsfromamorecompletebasissetc alculation.Thebest resultsareshowninFigure 4-1 andacomprehensivelistofresultsissummarizedin Table 4-2 andTable 4-3 ItshouldbenotedthatforbothEOMandSTEOMcalculationsth emeanaverage deviationisontheorderof0.1eV.However,theerrorinthec alculatedabsorption crosssectionissensitivetothespectralrange:thelowere nergywavelengthscarry ahighererrorthanthehigherenergywavelengthsasreecte dintheMADvaluesfor theenergiesinnanometerunits.Sincetherearenosignica ntdifferencesbetween STEOM-CCSDandEOM-CCSDmethodsfortheexcitedstatescons ideredinthiswork, EOM-CCSD/POLisusedinallsubsequentcalculationswithth enalenergiesofthe calculatedspectrumshiftedtothebestresultsfromTable 4-2 andTable 4-3 .Theenergy adjustmentsfortheabsorptionpeaksare:-21.8nmfortheAs tate,-9.9nmfortheB stateand-9.4nmfortheCstate. 54

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Thereisnoreasontosupposethatdoublyexcitedstatesares ignicantcontributors sincetheaverageexcitationlevel(AEL)[ 148 ]isnotgreaterthan1.07inanyofthe EOM-CCSDcalculationsfortheexcitedstatesconsidered.4.1.2AbsorptionCrossSectionModel Thisportionoftheresultssectionwillillustratethevali dityofassumptionsmadein themethodologysectionusingNaOHasanexample.4.1.2.1Dissociativecoordinateastheprimarycoordinate Figure 4-2 showsthebehaviorofthelowenergyexcitedstatesasafunct ionof distancebetweenthesodiumandoxygenatoms.Elevensingle pointcalculations areperformedforthedistancesrangingfrom1.7to5.0.Th eenergysurfacesare computedusingCCSDandEOM-CCSDmethodswitharestrictedH artree-Fock(RHF) referenceinthevicinityoftheequilibriumgeometryandan unrestrictedHartree-Fock (UHF)referencepasttheNa-Odistanceof2.5atwhichthebo ndbeginstobreak. ThevalencestateispurelydissociativeandtheRydbergsta tesareweaklybound. ThegroundstatesurfaceistwithaMorsepotentialwiththe unboundlimitsettothe sumofthecalculatedenergiesofNaandOHradicals(shownas apointat6.0in Figure 4-2 ).TheexcitedstatesurfacesarettedwithEquation 4–1 whichallowsa goodtforafullydissociativesurface,aswellasforaweak lyboundsurfacebysetting parameterBtozero.V ( x ) = A + Bx3 + C exp(Dx )(4–1) Thedipoletransitionmomentsaretusinga6 th orderpolynomial.TheR 2valuesof thetsaregreaterthan0.998withtheworsttbeingtheRydb ergstatesduetoasmall energygapattheUHF-RHFjunctionontheenergysurfaces.Th eUHF-RHFenergygap atNa-Odistanceof2.5forthegroundstatePESis0.23mH,fo rtheexcitedstatesAit is4.2mH,andforstatesBandCitis7.2mH. 55

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nr r nnr nr nrrrr nrrrr nrrrr nr nr nr nr nr Figure4-2.EnergiesanddipoletransitionmomentsofNaOHa longitsdissociative coordinate.SolidlinesforRHFsolutionsanddashedlinesa reUHF solutions.Thepointsat6.0representthetotalenergieso fthedissociated products.Thearenosurfacecrossingsobserved. 56

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4.1.2.2Computetheabsorptioncrosssectionoftheprimary mode Atthispoint,azerothorderapproximationtothespectrumm aybedetermined. Sincetheprimarymodechosenwasadissociativemode,theth eoreticalspectrumwill havetheformofacontinuumcrosssectioncenteredatthever ticalexcitationofthe equilibriumgeometryaugmentedbythedipoletransitionmo mentintensity.Onecanalso makeadifferentchoicefortheprimarymodeandletitbetheb endingcoordinate,in whichcasethespectrumwillhavenestructureofthevibron icprogression.Sincethe experimentwasnotdoneatahighenoughresolutiontoyieldv ibronicpeaksandweare interestedinphotodissociation;thechoiceofprimarymod eremainstheNa-OHstretch. Usingtheabove-mentionedttingparametersfortheenergy anddipolestrength surfacesthecrosssectioniscalculatedusingaDVRprogram implementedinFORTRAN specicallyforthispurpose.Theresultingcrosssectioni sshowninFigure 4-3 TheenergyadjustmentsfromPOLtotheWMRbasissetareasfol low:A(-21.8 nm),B(-9.9nm),andC(-9.4nm).Theseenergieswillbeusedf ortherestofthedata. 4.1.2.3Considersignicantlythermallypopulatedvibrat ions Thecalculatedabsorptioncrosssectioncomingfromthepri marymodealone isinfairlygoodagreementwiththeexperimentallyobtaine dcrosssection.Itmaybe sufcientforthismolecule.Tofurtherimprovetheresults weneedtolookattheother vibrationaldegreesoffreedomandwhat(ifany)effectthey haveonthecrosssection. Includingtheeffectsofothervibrationalmodestothediss ociativemodewill broadentheenergyrangeofeachpeakaswellasmaketheinten sitiesmorerepresentative ofwhatisexperimentallyobserved.Forexample,thisstepw ouldbevitalingettingan accurateabsorptioncrosssectionforaforbiddenexciteds tatewhichisvibronically allowed. Vibrationalenergiescanbefoundintwoways:1)harmonicfr equenciesare calculatedusingACESIIand2)anharmonicfrequenciesared eterminedfromDVR. 57

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nr n n n n nnr nrrr n nnrnnrnnr Figure4-3.ThesecrosssectionsareobtainedfromtheFCove rlapbetweentheZPE vibrationalwavefunctionofthegroundstateandthedissoc iativecontinuum oftheexitedstatesweighedbythedipoletransitionmoment .Thesurfaces arecomputedattheEOM-CCSD/POLleveloftheoryandtheresu ltingcross sectionsareshownasdashedlines.Onlytheenergiesareadj ustedtothe bestcalculatedvalues. Table4-4.Characteristicsofthevibrationalfrequencies ofNaOH. ModeSymmetryTypeCCSD/POL( cm1 )DVR( cm1 )Exp.( cm1 )a Comments 1Bend272.31284.7300Include. 2Na-OStr.559.83526.6540Primarymode. 3O-HStr.3969.79–3637Exclude. Thesevaluesareconsistentwitheachotheraswellaswithex perimentaldataaslisted onTable 4-4 Thebendingmodeistheonlylowenergymodeinthismoleculea nditwillbe consideredinthenextsteps.TheOHstretchingmodeistoohi ghinenergytobe signicantlypopulatedandhaveanimpactontheabsorption crosssectionsoitwill notbeincluded.Thestretchingmodehasconsiderableanhar moniccharactersothat theMorsepotentialttothegroundstatePESprovidesbette ragreementwiththe 58

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experimentalfrequencythanwiththe abinitio result.Thebendingmodeistwitha quarticpotential.Thedifferenceinvibrationalenergyfr equenciesbetweenthe abinito andtheDVRvaluesis35wavenumbersforthestretchingmodew hichisattributedto anharmonicityofthePES.Thebendingmodeismuchbetterdes cribedbyaharmonic potentialyieldsamuchsmallererrorbetweenthe abinitio andtheDVRvalues.The bendingmodewastwithaquarticfunction.Energyerrorsfo rvibrationalmodesonthe orderof50wavenumersintroducelessthan0.5nmerrortothe absorptioncrosssection intheworstcasescenario.4.1.2.4Determinetheimpactofeachsecondarymode IncaseofNaOH,thebendingmodeistheonlyoneselectedtobe inthesecondary-mode set.Forlargermolecules,therewillbemorelow-energyvib rationaldegreesoffreedom toconsidersofurtheranalysisofnormalmodesisneededtow eedoutonlythe importantdegreesoffreedom.Inclusionofvibrationalmod eswhichstronglycouple totheexcitedstatepotentialenergysurfacesisvitalfora naccurateabsorptioncross section.Emphasisisplacedonthescreeningbeingbothquic kandaccuratetominimize thenumberof abinitio calculationsthatneedtobeperformed.Thisinvolvespicki ng eachmodeinthesecondsetandcalculatingafewkeypointsal ongitssurface.Foreach setofpointstheimpactofthemoleculargeometrydeformati onalongaparticularmode ontheexcitationenergyandthedipoletransitionmomentis determined. Thenumberofactualcalculationscanbeoptimallyminimize dtoafewkeypoints onasurface: cis and trans conformersfordihedralrotationmodes,a+/-30degree deformationineachdirectionisappropriateforbendingmo desand+/-0.5for stretchingmodes.Sincethisapproachisstillinthetestin gstage,theaboveparameters arenottobetakenasabsolutesbutasastaringpoint.Symmet ryandbreakingthereof willneedsomeattentionpaidtoitsincesymmetricmodesand asymmetricmodes willhaveslightlydifferentbehaviors.Furthermore,ther earespecialvibrationalmodes 59

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Table4-5.Theimpactofthebendingnormalmodehasontheexc itedstatepotential energysurfacesandthedipoletransitionmoments. ExcitedStateABCA'B'Na-O-HAngle EE(nm)325.8237.8232.1349.7245.9120EE(nm)344.5239.3227.4344.5239.3180EE(nm)325.8237.8232.1349.7245.9240 Impact(nm)10.800.862.713.003.81 DTM(a.u.)0.660.030.130.340.02120DTM(a.u.)0.300.050.190.300.05180DTM(a.u.)0.660.030.130.340.02240 Impact(a.u.)0.690.230.180.080.35 thatneedtobetakenintoconsiderationsuchtheumbrellamo deofammoniaorthe ring-breathingmodeofbenzene. Therststepistogetalistofvibrationswhichhavethemost impactandtoexclude theonesthathavelittleornoeffect.Ifthereisareasonand capabilitytodomorepoints onthesurfaceswhichmattermostthencomputertimemaybeal locatedmoreefciently togetbettersurfacesformodeswithhaveagreaterimpact.T able 4-5 showshowthis procedureworksforthebendingmodeofNaOH. ItisclearfromTable 4-5 thattheerrorinenergyofthezerothordercrosssection (Figure 4-3 )isnograterthan10.8nm.Thelowerenergyvalenceexciteds tateismost affectedbythebendingmodeandtheerrorsintheRydbergsta tesaremoretolerable. ThedipoletransitionmomentforAandBstatesissignicant lyaffectedbythebending modewhichwouldhaveimpactonintensities.Figure 4-4 showstheenergysurfaces andthedipoletransitionmomentsurfacesasafunctionofth ebendingmodewhile keepingtheR NaOatitequilibriumvalueandrelaxingtheR OH. Forasmallmoleculesuchassodiumhydroxideitisinexpensi vetocomputeseveral morepointsalongthebendingmode,butisnotnecessarytodo sinceweareonly interestedintherangeofthepotentialswhichareclosetot heequilibrium.Ifahigh temperaturespectrumwereofinterestwhereasignicantfr actionofhighervibrational levelsarepopulated,thenitwouldmakesensetohavemorepo ints.Theagreement 60

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nr nn n nr n nnr Figure4-4.EnergiesanddipoletransitionmomentsofNaOHa longitsbending coordinate.Thepointswhicharelistedinthetablearethet hreepoints encompassedbytheverticallines.Severalmorepointswere donefor completenessbutwerenotnecessaryforaccuracy. 61

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Figure4-5.Thisisacartoonofavibrationalwavefunctioni napotential(inthiscasethe potentialofthebendingmode).Thechangeinthesurfaceene rgybetween theequilibriumgeometryandthegeometryofthehalf-maxim umiswhatis addedontothezerothorderdissociativepotentialfortheg roundandexcited states.Thedipolestrengthpotentialismultipliedbythef ractionoftheDTM HalfMax=DTM Max. betweentheDVRvibrationalenergyforthismodeandthe abinitio frequencysuggests thatthreepointsprovideagoodenought.Alargediscrepan cybetweentheenergy valueswouldbeasignaltodomorepointsforabetterpotenti al. 4.1.2.5Addtheeffectsofotherdegreesoffreedom Thettingparametersoftheenergysurfacesandthedipolet ransitionmoments forexcitedandgroundstatesareusedtoperturbthezerothorderdissociativestate andrecalculateFrank-Condonoverlaps.Thiscalculationi sdoneattwopointsas illustratedinFigure 4-5 :theoriginalequilibriumgeometryoverlap(whichcorresp onds tothemaximumofthevibrationalwavefunction)isgivenawe ightfactorof0.50anda weightfactorof0.25isassignedtothetwohalf-maximumpoi nts.Duetothesymmetry inthebend,thisdistributionsimpliesto50%oftheintens itycomingfromthepointat themaximumandanother50%oftheintensitycomingfromoneo fthepointsatthe half-maximum. DuetothebreakingoftheelectronicdegeneracyforstatesA andB,thedistribution is50%oftheintensitycomingfromthepointatthemaximum,2 5%oftheintensity 62

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Table4-6.OscillatorstrengthscalculatedfromtheDVRcro sssections. StateSinglePointZPEZPE+P1Convolutedfexp.(200K)fexp.(300K) A0.0540.0510.0510.0510.0170.023B0.0920.0950.0840.1080.0600.106 C0.0260.0270.024 Figure4-6.Theconvolutedcrosssectionsareshowninthick lines.Thisisthe1 st order spectrum. comingfromoneofthepointsatthehalf-maximumfromonesym metry(AandB)and another25%oftheintensitycomingfromoneofthepointsfro mtheothersymmetry(A' andB').Finally,thepeaksareconvolutedintoonecrosssec tionforeachexcitedstate asshowninFigure 4-6 .Theresultingconvolutedspectrumiscorrectinthe1 st order. Takingmorepoints:66%/33%Maxwouldbe2 nd order,75%/50%/25%wouldbe3 rd and soforth. 63

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4.1.2.6Temperatureeffectscanbeintroducedatthisstage ThespectruminFigure 4-6 doesnothaveanytemperatureeffectssinceonlythe rstvibrationallevelsarepopulated.Temperatureeffect smaybeaddedbyputting quantaintohighervibrationallevelsandrecalculatingth eweights.Noextratimeis requiredforthisasallFrankCondonintegralsarecalculat edatonce.At200Kthe contributionfromthesecondvibrationalleveloftheprima ryandthesecondarymodes totheabsorptioncrosssectionisonlyabout10%ofthetotal intensitywhichisnotvery noticeable.Onlywhenthetemperatureusedindeterminingt heBoltzmanndistributionis raisedabove500Kdoesanysignicantchangeinpeakintensi tyoccur. Figure4-7.Theconvolutedcrosssectionisshownwiththeex perimentalcrosssection Accordingto SelfandPlane thesignicantincreaseinintensityforthepeakat 220nmisduetotheincreaseintheFrank-Condonoverlap.Thi sassertionisbased onthefactthattheirCIS/6-311+G(2d,p)calculationofthe geometriesofgroundstate andexcitedstateshowedthatwhilethegroundstateinlinea r,theexcitedstateisbent. 64

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Inthepresentwork,thegroundstateandexcitedstatepoten tialenergysurfacesin Figure 4-4 showalackofatruedoubleminimumontheexcitedstatePES 1 which meansthatalineargeometryispreferredfortheexcitedsta tesintheenergyrangeof theabsorptioncrosssection.Furthermore,thedipolestre ngthforexcitationsBandC tendstodiminishasafunctionofangletherebyreducingthe totalintensity(asopposed toincreasingit).Thisisinsupportoftheconclusionthath otbandsoriginatingfrom populatingthebendingmodewillnotleadtoasignicantinc reaseinintensityforBand Cstates,butpossiblyfortheAstate.Thehigherintensityc rosssectionobservedby Self andPlane at300Kmaynotbeduetoatemperaturedependencebuttothefa ctthatitis ahigherresolutionexperimentandslightlymoreaccurate. TheconsistencycheckdescribedintheMethodssectionisdo neandresults presentedinTable 4-6 .Itappearsthatasthepeaksgetbroadened,theintegrated oscillatorstrengthsbecomeslightlylowerbutnomajordis crepanciesarenoted.The naltheoreticalspectrumispresentedinFigure 4-7 4.2Water:exampleofaboundsystem Intheprevioussection,itwasshownthatthefrequencypreselectionincombination withtheDVRvibronicspectrumprovidesaverygoodrepresen tationofaphotodissociative absorptioncrosssection.Inthissection,Iwillshowthatt hisapproachalsoworksona boundabsorptioncrosssection. Thewatermoleculehasthreevibrationalfrequencies:symm etricstretch(3657 cm -1 ),bend(1595cm -1 )andasymmetricstretch(3756cm -1 );thereferenceexperimental geometryisR OH = 0.958and \HOH = 104.478[ 62 ].Usingtheexperimental 1 TheenergysurfacesinFigure 4-4 werecalculatedwiththeconstraintthatR NaOisat itequilibrium.ThesesurfacewererecomputedsuchthatR NaOandR OHwereallowedto relaxandnodoubleminimumwasfound 65

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Figure4-8.AcollectionofseveralexperimentalVUVabsorp tioncrosssectionsfor water:Watanabe etal. [ 156 ],Yoshino etal. [ 163 ]andMota etal. [ 111 ].The theoreticalstickspectrumforseveralselectedbasissets areshown. geometrythephotoelectronicspectrumcomputedwithEOM-C CSDisshownin Figure 4-8 alongwithseveralexperimentalabsorptioncrosssections Thedecompositionoftheexperimentalspectraisasfollows .Thereisnodisagreement aboutthepeakatC,itisa1 B 1 ( b 1!3 pb 2 )allowedexcitation.Thecalculated verticalexcitationcorrespondstothepeakatAwhichisano therallowedexcitation1 A 1 (3 a 1!3 sa 1 ).ThisexcitedstateoverlapswiththepeakatBwhichcorresp ondsto anelectronicallyforbiddenexcitedstate(vibronicallya llowed)1 A 2 ( b 1!3 pb 2 )[ 26 89 ]. Table 4-7 showstheHartree-Fockorbitalenergiesandsymmetrylabel sfromWMR basissettohelpfollowtheexcitedstateassignments. 66

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Table4-7.MolecularorbitalsofwaterHF/WMR. Energy(a.u.)Energy(eV)Label -20.5632996227-559.55608293011a1 -1.3544751515-36.85715931292a2-0.7202241618-19.5983046595b2-0.5853340530-15.92775653373a1-0.5107691418-13.8987412302b1 0.03200453680.87088811613sa10.05007582451.36263307613pb2 Unfortunately,it'snoteasytounambiguouslyde-convolut ethesepeaksandgeta preciseoscillatorstrengthvaluesinceaportionoftheint ensityfromthe1 A 1excitedstate beasignicantportionoftheintensityofthe1 A 2state. Thevibronicspectrumat500Kiscalculatedforthe1 A 2statetoshowthe dipole-forbiddenstatebecomevibronicallyallowedasade monstrationoftheapproach describedinChapter3.Thebendingmodeischosenastheprim arymodeandthe OH-stretchischosenasaperturbation.Thevibronicspectr umisshowninFigure 4-9 Thespectrallinesshowamatchingbroadnessandpeakcharac teristicsasthe experimentalcrosssection. 67

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Figure4-9.Boldlines:OHstretchvibrationallevels,v str = 0(blue),v str = 1(green).Fine lines:vibrationallevelsatv str = 0andv str = 1.Theexperimentalspectrumis byMota etal. [ 111 ]. 68

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CHAPTER5 SINGLEREFERENCECCFORMULTI-REFERENCEPROBLEMS 5.1Background Traditionalhighordersinglereferencecoupledclusterth eory(CCSDT,CCSDTQ, CCSDTQP,etc.)[ 3 – 5 8 – 10 27 28 84 123 131 ]cansuccessfullysolvemulti-reference problemssuchasbondbreakingatasufcientlevelofapprox imation.Forarecentand comprehensivereviewofsingle-referenceandmulti-refer encemethodsusedforthe descriptionofbondbreakingsee[ 5 38 99 126 ].Therecoveryofthecorrelationenergy usingcoupledcluster(CC)methodsismuchmorerapidthanth oseofconguration interaction(CI)duetotheexponentialnatureoftheCCwave function.However, includingallhigherorderexcitationsisnotthesolution, becauseofcomputational demands.Hence,stepshavebeentakentolimitthemostexpen siveproceduresto asmallorbitalspacesuchasisdoneinCCSDt'q'andwithaddi tionofintermediate indexesinCCSDtq[ 124 125 ];aswellaswithTailored-CCmethod(TCCSD)[ 59 71 ] methods;thelaterbeingrelatedtotheRMRapproachofLiand Paldus[ 87 ]. 5.2Tailored-CCextension ThemethodsdescribedbelowaredepictedpictoriallyinFig ure 5-1 whereitiseasy toseethemaindifferencesamongthem. AsinglebonddissociationrelativetoanRHFreferenceisco nsideredatypeof multi-referenceproblemwherethesignicantdeterminant soriginateinthe spaceof thetwoelectron,twoorbital([2,2])space.TheTCCSDschem eusestheFCIcoefcients fortheactivespaceproblemandimposesthemontoafullspac eCCSDcalculation: jTCC i=je (ext^ T 1 +ext^ T 2 ) e (FCI^ T 1 +FCI^ T 2 )j 0i (5–1) Computationally,theTailored-CCwavefunctionisobtaine dintwoparts:rstthe activespaceamplitudeslabeledwithfciaredetermined jFCI i=je (FCI^ T 1 +FCI^ T 2 )j 0i (5–2) 69

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then,Equation 5–1 issolvedfortheexternalamplitudeswhilekeepingtheampl itudes obtainedfromEquation 5–2 constant.TheFCIfora[2,2]problemisequivalenttoa CCSDoraCISDcalculation.IfaCCSDcalculationisdoneinth eactivespace,the twot-amplitudes(t 1{ },andt 2{ })aredirectlyusedinthesubsequentrestricted CCSDcalculation;ifaCISDcalculationisdoneintheactive space,thet-amplitudes arederivedfromCIcoefcients via theclusterdecompositionanalysis:T 1 = C 1andT 2 = C 2C 2 1=2.Eitherwayisacceptabletoobtaintheactivespacet-ampli tudes. Thedecouplingbetweentheactive[2,2]spaceandthefullsp acecanbepathological intheTCCSDmethodwhichbecomesoneofthemajordrawbackso fanotherwise efcientwayofintroducinghigherordersofcorrelation.T hisdecouplingmanifests itselfinalargenon-parallelismerror(NPE),measuredast hedifferencebetweenthe maximumandtheminimumerroralongthepotentialenergysur face. Oursolutionistointroducesomeadditionalcouplingbetwe enthe[2,2]active spaceandthefullspace(seeFigure 5-1 regardingthediscussionbelow).Thisisdone withaCCSDTcalculationinanextendedactivespaceorwhati sknownasaCCsdt calculation.Theamplitudesassociatedwitha[2,2]determ inantaresavedfromthat CCsdtcalculationandthefullspaceCCSDequationsareallo wedtorelaxwhilekeeping theamplitudesfromtheCCsdtconstant.Thisapproachiscal ledXTCCSDfor"eXtended TCCSD".Inordertodetermineactivespaceindependenterro rsassociatedwiththe Tailored-CCSDprocedure,weusetheentireorbitalspaceto dotheCCSDTcalculation fromwhichthe[2,2]-relatedt-amplitudesaretaken( XTCCSD).Thiscalculationalso providestheCCSDTvaluesasareference. Anotherwaytousetheextendedactivespaceistosaveallthe amplitudesfromthe CCsdtcalculationandusethemtotailorthesubsequentfull spaceCCSDcalculation. ThiswillleadtoaverydifferentNPEandrelativeenergyerr orsincenowtheabsolute energyvaluewillbeclosertothereferenceCCSDTvalue.Wec allthisFXTCCSDfor "FullyeXtendedTCCSD"Thecalculationtimetendstobeslig htlyshorterthanthatof 70

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Figure5-1.Acartoonrepresentationofthemethodsused.Th etraditionalTailored-CCis ontheleftwithasinglebondactivespaceindicatedbythe and orbitals. TheextendedactivespaceisindicatedforXTCCSDandFXTCCS D methods.InXTCCSD,theactivespaceamplitudesaresavedbu tare calculatedwithaCCsdtcalculationintheextendedspace.I nFXTCCSD,the entiresetofamplitudesfromtheextendedspaceissaved. XTCCSDbecausetherearefeweramplitudestoconvergeinthe secondstep,usually takingfeweriterations. ThemainadvantageoftheFXTCCSDapproachiswhentheactive spaceis notnecessarilywelldened apriori .Unlikeintheclearcaseof -bonddissociation, sometimestheuserdoesnotknowwhichofthedeterminantsco nstitutethemulti-reference problem.Thisisoneofthegeneraldrawbacksofactivespace methods,especiallyfor difcultand/orunfamiliarmolecules,sincewhattendstoh appenistheactivespace isvariedbytheuseruntilsomeagreementwithexperimentmi ghtbereached.We believethatthisisnotwhatthefunctionofapredictivethe oryoughttobeandinsist thatthechoiceofactivespaceshouldbebasedonnumericalr esultsaloneandnot onanychemicalintuitionsoasnottocreateinherentlybias edresults.Tothatend, weproposeanautomatedactivespacedeterminingalgorithm (ASDA)whichndsa setofappropriateextendedactivespacesbasedonstatisti calanalysisoftheorbital energiesofthereferenceusedbythecoupledclustercalcul ations.Theusercaneither pickanextendedactivespacefromtheASDAsetorletthesele ctionprocessdoso automatically. 71

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Therewereseveralattemptstoautomateselectionofactive spacespublishedprior tothisworkwhichincludeanadaptivecoupledclusterschem e[ 97 ],aschemebasedon overlapintegralsofthereference[ 76 140 ]andaparticularorbitalcontributions(POC) analysis[ 98 ]. 5.3Extendedspacechoice ThemainfunctionoftheASDAistondtherst"big"energyga pinthevirtual(or occupied)orbitalswiththelogicbeingthatafterthatgapt hedenominatorslosemost oftheirsignicancebybecomingtoolarge.Soallsignican ttransitionsaretreated withahigherleveloftheory(CCSDT)andalltheotheroneswi thalowerleveloftheory (CCSD). Thefollowingdiscussiondetailsanautomatedwayofpickin ganactivespacefor eithertheextendedtailoredcoupledclustermethod(XTCCS D)orthefullyextended tailoredcoupledclustermethod(FXTCCSD)bothofwhichare describedintheprevious section. Agenericamplitudemaybedescribedas:T v o = tv o v(5–3) Wheretisacoupledclusteramplitudecoefcient,viseithertheFockor2-electron integral, oareorbitalenergycontributionsfromtheoccupiedspacean d vareorbital energycontributionsfromthevirtualspace.Thegenericam plitudedoesnotdistinguish excitationlevel: e.g. o =i +jand v =b +bforaselectedt ab ij.Sincetheorbital energydifferenceappearsinthedenominator,weconcludet hatthemorepositivethe valueof vis,thelessthatamplitudetermwillcontributetothetotal groundstateenergy sinceas o v!1 ,T!0.Duetothisproperty,itiscommonpracticetoexcludecore orbitalsfromthecoupledclusterandcongurationinterac tioncalculations. TheextendedactivespaceasusedbytheXTCCSDandFXTCCSDme thods needstobeestablishedsuchthatthecutoffisinthemid-val ence,whichislessobvious 72

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thanasimplecoreorbitalexclusion.Thisisespeciallysoi nthevirtualspacewhere energylevelsmaybecloselypacked.Weemployastatistical analysisofthedistribution ofenergyvaluesofvirtualorbitalstodeterminethebestpl acesfortheactivespace boundary. TheexamplesintheResultssectionillustratenumerically howeachofthesteps areaccomplishedanditispertinenttosaythatwhiletheref erenceorbitalshereare Hartree-Fock,itisnotanecessity.Thestatisticalmodeld rawsfromtheavailablesetof orbitalssotheonlyconditionforsuccessisthattherebeen oughofthemtopresenta meaningfuldistribution. TheHartree-Fockorbitalsareseparatedintooccupiedandv irtualsets.Forthe moleculesstudiedhere,therearenotenoughoccupiedorbit alstoofferameaningful distributionsoonlythecoreorbitalsareexcludedfromthe extendedactivespaceforthe occupiedblock.Thevirtualorbitalsaresortedinorderofi ncreasingenergyvaluessuch thattheorbitalenergymaybeexpressedasafunctionoforbi talnumber,E ( n ). Thenumericalderivativeoftheorderedorbitalenergyisth en:dE ( n ) dn = E ( n + 1)E ( n ) ( n + 1)n(5–4) Sincednisalwaysone(thoughforamoleculewiththousandsofbasisf unctions,we maywanttomakeitgreaterthanone),Equation 5–4 becomes:dE ( n ) = E ( n + 1)E ( n )(5–5) Thesignicanceofthecontributioncontainingthen thorbitalismeasuredbythe reciprocaloftheenergyR ( n ) = dE ( n )1(5–6) wherethelargerthevalueforR ( n ),themoresignicantisthecontributionfrom denominatorscontainingthen thorbital.Coreorbitalsareexcludedfromfurtheranalysis. 73

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ThedistributionofR ( n )isexponential,thereforetoachieveanormaldistribution ,R ( n )is transformedtoln ( R ( n )). Thekerneldensityestimation(KDE)andquantile-quantile (QQ)plotsshowthatthe distributionoftheln ( R ( n ))datapointsisclosetoanormaldistribution,butnotalways unimodal,especiallyatpointsawayfromequilibrium.Havi ngmultiplemodesatthe dissociationlimitisnotsurprisingconsideringthatthee nergylevelsofatomicorbitals areinherentlybi-modal(n-levels(largeenergygaps)andn -manifoldlevels(smallenergy gaps)).Atequilibriumthemolecularorbitalsarecoupledt hroughthelinearcombination ofatomicorbitalsprocedure,andthustendtoproducesmoot herdistributions.Ifmultiple modesaredetected,theASDAwillonlyconsiderthemodewhic hsamplesthebiggest gapsinthedistribution. Theln ( R ( n ))dataarettoeitheraunimodalormultimodalnormaldistrib ution andthemean( )andthestandarddeviation( )computed.Therangeofinterestis U = fortheupper-boundcutofftotheactivespace.The Uisthentransformed backtotheenergydomaindE U = exp( U ).Then th + 1orbitalofanydE ( n )>dE Uis savedtoalistasacontenderformarkingtheactivespacecut off. ThenaloutputfromASDAisalistoforbitalswhichfulllth econditionofbeingon theuppersideofasignicantly(bymorethanonestandardde viation)largeenergygap. Theprogramwillattempttogowiththerstcutofforbitalon thelistandchecktomake surethatatleast20%ofthetotalnumberofbasisfunctionsi sincludedandthatalllarge amplitudesfromasecondordermany-bodycalculation(MBPT (2))arewithintheactive space.Ifanyoftheaboveconditionsfail,theactivespacei sexpandedtothenextcutoff candidateonthelistandthechecksarerepeated.Thesechec ksareanunfortunate consequenceofbasissetcontractionswhichcanleadtoearl ygapsinthevirtualspace energymanifold. Atthispointtheactivespacecalculationwillproceedasde scribedintheprevious subsection.Dependingonthepurposeofthecalculation,th eusercanalwaysintervene 74

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andchoosewhichofthepossiblecutoffstouse.Forexample, ifthenalgoalisto generateaPES,inordertominimizediscontinuitiesonecan chooseacutoffwhichis commontoallthepointsonthePESwhichwewillrefertoasani ntersection( \ )active space.ThischoiceleadstobetterNPEbutcanbemoreexpensi ve.Ifthegoalisto determinetheenergydifferencebetweentwopoints,onecan chooseaconsistentactive space(likeinaPESgeneration)oronecanlettheASDAchoose theminimalcutoff space.Bothchoicesareevaluatedintheresultssectionand accuracyisweightedby computationaladvantage.Inanycase,wehighlydiscourage pickingacutofforbital whichisnotontheASDAoutputlistjusttosatisfyactivespa cesizeconsistency. 5.4Statisticalanalysistechniques Thissectioncontainsthebackgroundnecessarytoundersta ndthestatistical analysisperformedbyASDA.5.4.1Quantile-quantileplot AQuantile-QuantilePlot,orQQPlot,isaplotcomparingthe probabilitydistribution oftwosetsofdata.Quantilesarepointstakenatregularint ervalsfromadatasample, themostfamiliarofwhichisthe2-quantileorthemedian. Generally,aQQPlotwillcomparethedistributionofadatas etwithatheoretical distribution(normal,gamma,Poisson,geometric,etc.).I ftheplotformsastraightline withaslopeof1andinterceptof0,thenwecanconcludethatt hedatafollowswhatever theoreticaldistributionitwascomparedwith. ItiseasytoseeoutliersonaQQPlotandanyotherpeculiarit iesofthedatasuchas ifitfollowsacombinationofprobabilitydistributions.T hemainreasonforevaluatingthe QQplotistomakesurethatthestandarddeviationwhichisus edasacutoffparameter iscalculatedforthecorrectprobabilitydistributionoft hedata. 5.4.2Kerneldensityestimation Thefollowingexplanationisadaptedfromabookcalled DensityEstimation by B.W.Silverman[ 95 ]. 75

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Adatasetcanbepresentedasahistogram.Abetterrepresent ationofthisdatais todetermineauniformdistributionttoit.Toaccomplisht hat,wedeneakernelK: Z 1 1K ( x ) dx = 1(5–7) Inthecaseofnormallydistributeddata,thekernelusedisa normaldensity function.Theideaistoassignanormalprobabilitydensity function(aGaussian)to everydatapoint,X i.TheseGaussianfunctionsareconvolutedtoproducetheKDE plot via:KDE ( x ) = 1 nh nXi =1 K ( xX i h )(5–8) wherehisthewidthoftheGaussian,generallyreferredtoasasmoot hing parameterorbandwidth.Thebenetsincludecontinuityand theeaseofinterpretation ofthedatadistributionsinceKDEavoidsthejaggededgesof ahistogram.Forthe purposeofevaluatingtheresultsfromASDA,theKDEplotiss uperiortoahistogram plotbecauseitiseasytoseethemodalityofthedistributio n( i.e. howmanypeaks). 76

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CHAPTER6 ACTIVESPACECCEXAMPLES Inthisworkweexploretheerrorsthatarisefromthetraditi onalTCCSDandimprove uponitwhilekeepingthecomputationalcostmanageable.FH (hydrogenuoride)and F 2 (molecularuorine)arechosenasclassicexamplesofasing lebond-breakingMR problem. 6.1Hydrogenuoride TheFHexampleisusefulbecauseitsPESisverywelldocument edinthe literature[ 49 60 64 71 88 91 97 101 126 135 138 ]anditprovidesagood platformforerroranalysisofanynewmethod.Theexperimen taldissociationenergy isknown[ 23 31 165 ]andhasbeencalculatedtohighaccuracywiththequantum MonteCarlomethod[ 94 ].Unfortunately,mostoftheavailablecalculationsaredo newith verysmallbasissetsdueeithertocomputationallimitatio nsofthemethodsused,or tothelimitationoftheFCI[ 10 ]withwhichthecomparisonisusuallymade.Thiswork demonstratesthatlargerbasissetsmaycauseproblemswith convergenceandmake errorsthatmightbesmallwithdouble-zeta-typebasisgain signicance.Eventriple-zeta basissetsarenotadequatetoreacharesultwhichiscompara bletoexperiment. TheCCSDTmethodhasbeenshowntosuccessfullycomputethed issociative PESofasinglebond[ 5 ].Wefurthertestthismethod'sviabilitybycomparingseve ral pointsonthePEStotheFCImethodforFHwiththecc-pVDZbasi sset.Theenergy errorbetweentheCCSDTandtheFCImethodforFHatthedissoc iationlimit(3.20) is0.2mE handCCSDTisknowntorecovermorethan99%correlationenerg yat equilibrium. Figure 6-1 showsthedissociativePESenergyerrorwithrespecttoCCSD TofFH from0.0.917to3.20computedwithTCCSD,CASCCSD[ 64 ],CASCISD(MRCI)[ 20 122 ]andCCSD. 77

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Table 6.1 showsthedissociationenergyvaluescomputedwiththemeth ods describedabove.TheMRCIenergyliesslightlyhigherbutth eD evalueisacceptableso whiletheabsoluteenergyisabittoohigh,therelativeener gymayyetbetrustworthy. TheerroroftheCASCCSDmethodisclosetothatoftheCCSDint heequilibriumregion butitdoesnotdivergeastheCCSDdoes. Thereareseveralformulaswhichcanbeusedtoextrapolatec orrelationenergybut itisunclearwhichisappropriateforthemethodsbasedonTa ilored-CC.Furthermore, theerrorsoftheseextrapolationscanbelargerthantheerr orbetweenthelargestbasis setcalculationinthisworkandexperiment[ 41 ].Forexample,theCCSDTcorrelation energymaybeextrapolatedbyatleastthreewellknownmetho ds:E1 corr[T,Q,5] = a + bX3(6–1)E1 corr[T,Q,5] = a + bXc(6–2) wherefunctionparametersa,b,and/orcarettedtothecardinal[T,Q,5]numbers ofthecc-pVXZbasisset.Alsoatwo-pointextrapolationcan beusedusingthecc-pVQZ andcc-pV5Zenergies:E1 corr[Q,5] = 4 3Ecorr[4]5 3Ecorr[5] 4 35 3(6–3) Hatree-Fockenergiescanbeextrapolatedwiththeexponent ialdecayformula:E1 HF= a + bexp (cX )(6–4) UsingEquation 6–4 withEquations 6–1 6–2 ,or 6–3 ;theD evaluesforCCSDT are:227.72mE h,230.50mE h,or228.77mE hrespectively.Theconclusionisthatwhile theCCSDTprobablyovershootstheexperimentalreferenceb ysomesmallamount, itisunclearfromtheuncertaintyintheextrapolationform ulasbyhowmuch.Thisis consistentwiththeconclusionofHalkier etal.. [ 53 ]wherea( T Q 5)extrapolationcould 78

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0 5 10 15 20 25 11.522.53EnergyErrorvs.CCSDT(mH)R FH (Angstrom) FHGroundStateNPE CASCCSD CASCISD(MRCI) CCSD TCCSD Figure6-1.DissociativePESenergyerrorwithrespecttoCC SDTofFHfrom0.917to 3.20computedwith[2,2]FCI-Tailored-CCSD,CASCCSD,CASCISD(MRCI),andCCSD.Basissetiscc-pVDZ. have 3mE hoferrorandatwo-point( Q 5)extrapolationcouldhave 1.5mE hoferror. SeveralmE herrorsbetweendifferentextrapolationtechniquesisalso evidentinthe comprehensivebasissetextrapolationevaluationstudyof Feller elal .[ 41 ]Itispossible thatconventionalformulassuchastheoneslistedaboveare notsufcienttodescribe moleculessuchasFHorF 2 duetoratherpoorRHFreference.Sincetheunextrapolated CCSDTvalueis226.16mE hwhichisalreadywithinamE hdifferencefromthe experimentalandtheQuantumMonteCarlovalues,thereisno usefulinformationto begainedbydoingtheseapproximatebasissetextrapolatio ns. 79

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Table6-1.Dissociationenergyforhydrogenuoride.Energ iesare reportedinmE hunits. cc-pVDZcc-pVTZcc-pVQZcc-pV5Z CCSD222.55246.10252.45255.04 CCSDT201.68219.87224.41226.16 CASCCSD201.0 CASCISD(MRCI)200.6 TCCSD203.38221.38225.69227.45 XTCCSD202.22220.04224.51226.70 XTCCSD201.88219.10223.69225.65 FXTCCSD203.34220.61225.50226.89 experimentalvalues a 225.92 b 225.82 c QuantumMonteCarlo226.3 d a ExperimentalvaluesarereportedminustheatomicSOcoupli ngof -0.605580mE handrelativisticcorrectionsof-0.302784mE h[ 42 ] b FromRef.[ 23 ]; c FromRef.[ 31 165 ]; d FromRef.[ 94 ]. Table6-2.NPEforhydrogenuoride. EnergiesarereportedinmE hunits. cc-pVDZcc-pVTZcc-pVQZ TCCSD6.05.55.4 XTCCSD0.61.92.8 XTCCSD1.82.02.8 FXTCCSD a 1.70.91.1 a Forthecc-pVQZtheNPEisreportedforthe consistentactivespace,theNPEforminimalspaceis2.5mE h.Fortheotherbasissetsthe minimalactivespaceisalsotheconsistentactivespace. 6.1.1SourceofNPEinTCCSD TheNPEoftheTCCSDis 6mE hwhichislargerthantheerrorbetweenCCSD andCCSDTatequilibrium.Therearetwosourcesoferrorinth eTCCSDmethod:1) theerrorthatcomesfromthelackofcouplingbetweentheact iveandinactivespacein theFCIcalculation;2)theerrorthatcomesfromthefullspa ceCCSDcalculationwhich, basedontheerroraroundequilibrium,shouldbe 2mE h. 80

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-3 -2 -1 0 1 2 3 4 5 11.522.53EnergyErrorvs.CCSDT(mH)R FH (Angstrom) FHGroundStateNPE TCCSD XTCCSDXTCCSD FXCCSD CASCCSD Figure6-2.DissociativePESenergyerrorwithrespecttoCC SDTofFHfrom0.917to 3.20computedwithTCCSD, XTCCSD,XTCCSD,FXTCCSDand CASCCSD.Basissetiscc-pVDZ. Inordertoaddresstherstissue,wecomparethet 1{ },andt 2{ }fromthe [2,2]FCIcalculationtothet 1{ },andt 2{ }fromtheCCSDTinFigure 6-4 .There isalargedeviationbetweenthet-amplitudesfromthefullC CSDTcalculationandthat ofthe[2,2]FCIcalculationwhichsuggeststhattherearere gionsonthePESwhere thecouplingbetweenthe orbitalsandtheotherorbitalsissignicant.Toverify thatthisisthecausefortheNPE,weextractthet 1{ },andt 2{ }fromtheCCSDT calculationandfollowwiththerestrictedfullspaceCCSDc alculation(thesecondstep oftheTCCSDcalculation).TheNPEcurvesobtainedwith XTCCSDareincludedin Figures 6-2 and 6-3 81

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-2 0 2 4 6 8 10 11.522.53EnergyErrorvs.CCSDT(mH)R FH (Angstrom) FHGroundStateNPEBasisSetDependence cc-pVDZ cc-pVTZ cc-pVQZ TCCSD XTCCSDXTCCSD -3 -2 -1 0 1 2 3 11.522.53EnergyErrorvs.CCSDT(mH)R FH (Angstrom) FHGroundStateNPEforFXTCCSD cc-pVDZ cc-pVTZ cc-pVQZ Figure6-3.DissociativePESenergyerrorwithrespecttoCC SDTofFHfrom0.80to 3.20computedwithTCCSD, XTCCSD,XTCCSDandFXTCCSD.Basis setsarecc-pVXZ(X=2,3,4). 82

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-0.03 -0.02 -0.01 0 0.01 0.02 -1-0.8-0.6-0.4-0.200.2Errorvs.CCSDTt-amplitudesfromCCSDT CoupledClustert1 f ; g andt2 f ; g Amplitudes t2 f ; g t1 f ; g t1-[2,2]FCIt2-[2,2]FCI t1-CCsdtt2-CCsdt Figure6-4.AmplitudesfromthedissociativePEScalculati onsofFHfrom0.917to 3.20.Theerrorbetween[2,2]FCIamplitudes,CCsdtamplit udesand CCSDTamplitudesareplottedvstheCCSDTamplitudevalues. Basissetis cc-pVDZ. The XTCCSDmethodhassignicantlyreducedtheNPEandthemajori tyofthe remainingerror( 2mE h)isassociatedwithCCSD.Unfortunately,thisisnotavery cost-effectiveapproach.Itisclearthatinordertoreduce theNPE,somecorrelation betweenthe[2,2]spaceandtheextendedspacemustbeallowe dtooccurbutnotallof thatspaceneedberequired.Forthisreasonanextendedacti vespaceisintroduced. ItmustbesmallenoughtoallowatimelyexecutionoftheCCSD Tcalculationbutlarge enoughtocapturethecorrelationoftheorbitalswhichpart icipateinthe -bond.The methodwhichthusreducestheNPE via theextendedactivespaceisXTCCSD. 83

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InordertodeterminetheinuenceofbasissetchoiceonNPE, thePESforFH dissociationisalsocomputedwithacc-pVTZ,cc-pVQZ,andc c-pV5Zbasissets.The sametrendasshowninFigure 6-4 isobservedacrosseachbasisset.Lookingatthe NPEacrossthebasissetsinFigure 6-3 andTable 6.1 weseeadifferenceintrend betweentheTCCSDmethod(atthehighextremeofde-coupling )andthe XTCCSD method(atthelowextremeofde-coupling)withtheNPEchara cteroftheXTCCSD methodmuchclosertothatof XTCCSDthantoCCSD. 6.1.2FXTCCSD WhileXTCCSDhasdoneagoodjobofreducingNPEandobtaining theexperimentalD evalues,therecanbebenetstokeepingtheentireextendeds paceandnotjustthe orbitalsassociatedwiththe[2,2]problem.Therstreason istolowertheabsolute errorbetweenXTCCSDandCCSDTwhichismainlyduetotheerro rbetweenCCSD andCCSDT.Thesecondreasonistotreatafewotherlessimpor tantdeterminants atahigherleveloftheorybecauseitmaynotalwaysbeobviou swhichdeterminants makeuptheMRproblem.TheNPEcurvesobtainedwithFXTCCSDa reincludedin Figures 6-2 and 6-3 ThemaindrawbackoftheFXTCCSDmethodisthatitismoresens itivetotheactive spacechoicethantheXTCCSDmethod.ThemaximumerrorinXTC CSDduetoactive spacechoicecanbeestimatedbysimplylookingatthediffer encebetweenitandthe reference XTCCSDmethodwhichusestheentirespace.Theeffectsaremin orsince onlytwoamplitudesareaffected.InFXTCCSDtheexternalsp aceamplitudesrelax via CCSDwithrespecttothewholeextendedactivespaceamplitu desandsmalleffectson theenergy(notthemainMReffect)arecompounded.6.1.3Activespacechoice TheASDAasdescribedintheMethodssectionisusedtoperfor mstatistical analysisonthereferenceorbitalenergiestodeterminethe boundsoftheactivespace. 84

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SinceFHhassofewelectrons,onlytheonecoreorbitalisdro ppedfromtheoccupied spaceandASDAworksonthevirtualspace. Figure 6-5 showstheresultsforafewpointsalongthePES.Inaneffortt odealwith multimodaldistributions,theenergygapthresholdisincr ementallyincreased,untilall pointsbelongingtothemodewiththelargestenergygapsare isolated.Thisprogression canbeseeninthekerneldensityplotsofln ( R ( n ))asthedatasetgoesfromincluding allenergydifferences(pink)toincludingonlythelargeen ergydifferences(green).When theprogramdeterminesthatthedistributionisunimodal(b yanystandardmodalitytest suchasHartigan'sdiptest),itstopsincreasingthethresh old.Anotherapproachisto estimatethenumberofmodes,useamulti-modaldistributio ncurve-ttingprocedure suchasthemixtoolsRpackage,andonlyusetheresultsfromt hemodewiththelowest mean.Eitherapproachyieldsthecutoffsemployedinthisst udyandthoughsomesmall variationmaybepossible,nonehaveyetbeenseen.Theforme rapproachisusedin thisstudyasitismoreinagreementwiththephilosophyofmi nimizinguserintervention. TheQQplotsinFigure 6-5 showthatthedistributionisveryclosetobeingnormal andonecanevenseethedifferentmodesbylookingattheclus teringofthedataalong the1,1diagonal.TheoutliersatthehighervaluesoftheQQp lotsshowafeworbitals nearlydegeneratetoeachotherwithanenergydifferencesl ightlygreaterthan0.1mE hwhichistherstthresholdusedtodiscarddegenerateorbit als. AtypicaloutputfromASDAproducesalistofcandidateorbit alslikeso:++++++++ Possible Cutoff Orbitals at rFH= 1.30 Cutoff: 0.622599112125938 31 49 60 61 72 73 0.7435471 0.7033709 0.9927676 2.240253 0.9165902 1.311481 ++++++++ Possible Cutoff Orbitals at rFH= 1.4085

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-50510 0.00.20.40.60.81.0 KDE rFH= 0.917 ln(R(n))Probability Density -50510 -50510 QQ Plot rFH= 0.917 Normal DistributionDistribution of ln(R(n)) -50510 0.00.20.40.60.81.0 KDE rFH= 3.20 ln(R(n))Probability Density -50510 -50510 QQ Plot rFH= 3.20 Normal DistributionDistribution of ln(R(n)) Figure6-5.GraphicaloutputfromASDAisshownfortheequil ibriumanddissociation limitgeometriesofFHatcc-pVQZbasis.TheKDEplotofln ( R ( n ))isplotted fordatawhichincludesallenergydifferences(pink)andpr ogressestoward thedatasetwhichincludesonlythelargeenergydifference s(green).The QQplotsshowthattheprobabilitydistributionofln ( R ( n ))isnormal. 86

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Cutoff: 0.681974319543561 31 49 61 72 73 0.7040421 0.7261902 2.76404 0.8659455 1.416099 +++++++++Thisoutputshowsthatthelowestpossiblecutoffhappensat orbital31.Italsoshows thatthecutoffthresholdhasincreasedfrom0.6226Hto0.68 20Hgoingfrom1.30to 1.40pointsonthePES.Thelastlineisthesetofenergydiff erencesbetweenthe orbitallistedanditspredecessorsoitbecomesapparentth atwhileorbitals31,49, and60(at1.30)areslightlylargerthanthecutoffthresho ld;orbital61hasamuch largergap.Iftheuserwantstointerveneandpickorbital61 asthecutoffprovidedthe computationalcostismanageable,heorshecandosoatthist ime. Thedistributionpatternsdonotchangemuchfortheserieso fcc-pVXZbasissets sotheresultsshownhereforthecc-pVQZaretypical.Howeve r,pastthecc-pVTZbasis settherebeginstobeadifferencebetweentheminimalactiv espacecutoffandacutoff thatisconsistentwiththeentirePES.Generally,thisisdo minatedbythedifferenceof cutoffsatequilibriumandatthedissociationlimitasshow nintheFHcc-pV5Zexample:++++++++ Possible Cutoff Orbitals at rFH= 0.917 Average: 1.47006758809646 Cutoff: 0.701681769183348 60 68 83 104 107 108 114 118 121 124 125 138 142 145 0.9844741 0.7715318 0.9273001 0.7234571 0.8695963 1.001046 0.7500163 1.077662 0.8322482 0.868769 0.7360771 2.498539 1.192338 1.434257 ++++++++ Possible Cutoff Orbitals at rFH= 3.20 Average: 0.5372321321124487

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Cutoff: 1.71685675760544 108 111 125 141 2.905959 2.110185 4.525748 3.324365 +++++++++Here,theminimalcutoffwouldbe60atequilibriumand108at dissociationwhichleads toa0.55mE herrorwithCCSDTforD e.Anintersectioncutoffwouldrequirethatboth calculationsbeperformedwith108forcutoffleadingtoane rrorof0.71mE h;amarginal difference.ThesearesummarizedinTable 6.1.3 .Itisclearfromthistablethatthere isnogreatbenetforthecalculationofD etouseaconsistentactivespacefortheFH example.TheNPEisabitlargerfortheminimalactivespacec hoicesmainlyduetothe difcultmiddleportionofthePESwherealotofsurfacecros singsbegintohappenand solutionsmaybecomeunstable.However,forthetwoextremi ties,therelativeerrorto CCSDTisaboutthesameinbothcasesofactivespaceselectio n. Thereducedactivespacefractionofthetotalspacegeneral lybecomessmalleras thebasissetbecomeslarger.Becauseofthis,largerdiffer encesincomputationtime areobservedasthebasissetsizeincreasesbetweentheCCSD Tcalculationandthe Tailored-CCmethods.Table 6.1.3 summarizesthereductioninorbitalspaceaswell asthesubsequentspeed-upofthecalculationcomparedtoth eequivalentCCSDT calculation. TheXTCCSDmethodisnotnearlyassensitivetotheactivespa cechoicesothe minimumpossibleextendedspaceisalwaysused.Thetimings oftheXTCCSDarein thesamerangeastheonesforFXTCCSD.TheTCCSDmethodhasas peedupofabout 50xcomparedtoCCSDTforthecc-pVQZandcc-pV5Zbasiswhich reducesto8xfor thecc-pVTZbasis. 6.2Fluorinemolecule F 2 ischosenbecausetheUHFsolutionisnotboundandtheRHFsol utionisonly boundduetoitsincorrectseparation.Thereforeitprovide satestofasinglebond 88

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Table6-3.Orbitalspaceandperformanceforhydrogenuori de. cc-pVXZ a [e,o]FS[e,o]AS%ReducedSpeedupAbs.Err. cD eErr. c D[10,19][8,12]37– b 0.911.66 T[10,44][8,29]345x0.760.74Q \ [10,85][8,58]3218x1.071.09 Q[10,85][8,47]4545x1.091.085 \ [10,146][8,106]277x1.280.71 5[10,146][8,58]6024x1.160.55 a The \ superscriptstandsforresultsobtainedwiththelowestcom mon cutoff; b Thecalculationusedforthespeedbenchmarkistheenergy computationatequilibriumgeometry;c WithrespecttotheCCSDTreferencevaluesinmE hunits. dissociationpotentialenergysurface(PES)[ 21 40 60 88 91 126 135 138 ],and ismoredifculttocomputewithlargerbasissets.Thereisa lsoaverylargeerror inthedissociationenergyforCCSDwiththeRHFreferencebe ingnotoriouslybad. Again,mostoftheavailablecalculationsaredonewithvery smallbasissetswiththe exceptionofBytautas etal. [ 21 ]andEvangelista etal.. [ 39 ]whohavedoneaCBS extrapolationfortheF 2 PES.TheexperimentalvalueforthedissociationenergyofF 2 is wellestablished[ 62 162 ]. ThegroundstatedissociationPESofF 2 isanotoriouslydifcultproblemforsingle referencecoupledclustermethods.TheCCSDerrorinthedis sociationenergyisabout thesamemagnitudeastheactualdissociationenergy.Event hedissociationenergyof theCCSDTisknowntobeafewmE hhigherthantheFCIdependingonwhichbasisset isused. InTCCSD,themajorityoftheamplitudes(otherthanthe[2,2 ]dominantones) arecalculatedattheCCSDlevelbutdespitethatfact,theTa ilored-CCabsolute energyvaluesareclosertotheCCSDTvaluesthantheCCSDval uesalongthePES. Furthermore,therelativeenergy(D e)oftheextendedTailoredmethodsareinexcellent agreementwiththeCCSDTvalues. 89

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Table6-4.Dissociationenergyformolecularuorine.Ener giesarereportedinmE hunits. cc-pVDZcc-pVTZcc-pVQZcc-pV5Z CCSD84.07110.14119.90119.87 CCSDT43.4357.8461.0062.42TCCSD54.0267.3669.4869.33 XTCCSD45.9859.9162.6763.88 FXTCCSD44.3258.1560.8462.23 experimentalvalues62.56 a 62.13 b a FromREf.[ 21 162 ]; b FromHuberandHertzberg[ 62 ]minus atomicspin-orbitcouplingof-1.22705mE handrelativisticeffect of-0.031871mE h[ 42 ] Table 6-4 showsthecalculateddissociationenergyofF 2 forallbasissetsincluding theCBSvalues.Asthebasissetincreases,theD efromtheCCSDcalculationremains abouttwiceaslargeastheD efromtheTailored-CCmethodsandCCSDT.The TCCSDovershootstheexperimentalresultsbyabout 7mE hbutwiththeaddition ofactivespaceextension,thaterrorgoesdownto 2mE h.Basedontheconclusion ofBytautas etal.. andEvangelista,thereshouldbeasmall( 2mE h)errorinthe dissociationofF 2 whentreatedwithCCSDTduetotheverystrongmulti-referen ce characterofthisproblemthatcanbeamelioratedbytheaddi tionofquadruples, pentuples,etc.IntheCBSlimitinTable 6-4 theCCSDTcalculationisinverygood agreementwithexperimentalvaluesoncethespin-orbitcou plinghasbeensubtracted fromtheexperimentalvalues.Thereissomeroomforimprove mentoverCCSDTfor thepotentialenergycurveinthemiddlerange(2.20to3.00 )wherethereisstillan unphysical 1mE hbump,whichisindicativeofthemulti-referenceaspectsin theF 2 PES. Theerrorsbetweenthevariousmethodsaremoreapparentint heF 2 examplethan intheFHexampleduetoastrongermulti-referencecharacte raandlargernumber ofbasisfunctions.Theseerrorsareapparentinthewrongen ergyofCCSDaswell asinthestrugglingTCCSD.Inordertodeterminetheinuenc eofbasisset,theNPE 90

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Table6-5.NPEformolecularuorine.Energiesarereported inmE hunits. cc-pVDZcc-pVTZ TCCSD12.512.6 XTCCSD3.53.8 XTCCSD2.72.6 FXTCCSD1.30.5 withrespecttoCCSDTisplottedforthetwobasissetswherei tispracticaltocompute theCCSDTPES:cc-pVDZandcc-pVTZasshowninFigure 6-6 andinTable 6-5 .The basis-dependenttrendoftheNPEforthemolecularuorinec aseisobservedtobe similartothatofthehydrogenuoridecase.However,allof theerrorsareamplied: NPEaswellastheabsoluteerror.TheNPEfortheFXTCCSDmeth odisremarkably small. TheactivespaceselectionforF 2 isdoneinthesamefashionasfortheFH example.ThekerneldensityplotsinFigure 6-7 showthattherearealreadytwo welldenedmodesat2.0whichcanbeattributedtothechang ingoforbitalsfrom moleculartoatomiccharacter.Itiseasierseeninthisexam plethanintheprevious onebecausethereisonlyonetypeofatompresenthere.These twomodespersist allthewaythroughcompletedissociationat8.0.Asbefore ,inthecaseofmultiple modes,onlythemodewhichcontainsthedistributionofthel argerenergygapsisused todeterminethecutoff.Thedistributionofln ( R ( n ))isnormalaccordingtotheQQplots (notshownintheinterestofspace). Thereducedactivespacefractionofthetotalspaceismoref avorableforF 2than FHduehavinglargeratomsinvolved.Becauseofthis,someof thespeedupvalues inTable 6-6 reachtwoordersofmagnitude.Startingwiththecc-pVQZbas issetthe minimumextendedactivespacechoiceandanintersectionex tendedactivespace choicediverges;fortheequilibriumgeometryinparticula r.Thedissociationpointtends tohavethelargercutoffvaluesjustasintheFHexample. 91

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0 5 10 15 2345678EnergyErrorvs.CCSDT(mH)R FF (Angstrom) F 2 GroundStateNPEBasisSetDependence cc-pVDZ cc-pVTZ TCCSD XTCCSDXTCCSD -2 -1 0 1 2 3 2345678EnergyErrorvs.CCSDT(mH)R FF (Angstrom) F 2 GroundStateNPEforFXTCCSD cc-pVDZ cc-pVTZ Figure6-6.DissociativePESenergyerrorwithrespecttoCC SDTofF 2 from1.40to 8.00computedwithTCCSD, XTCCSD,andXTCCSD.Basissetsare cc-pVXZ(X=2,3). 92

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-5051015 0.00.20.40.60.81.0 KDE rFF= 1.412 ln(R(n))Probability Density -5051015 0.00.20.40.60.81.0 KDE rFF= 2.00 ln(R(n))Probability Density -5051015 0.00.20.40.60.81.0 KDE rFF= 8.00 ln(R(n))Probability Density Figure6-7.GraphicaloutputfromASDAisshownfortheequil ibrium,2.0,and dissociationlimitgeometriesofF 2 atcc-pVTZbasis.TheKDEplotofln ( R ( n ))isplottedfordatawhichincludesallenergydifferences(p ink)and progressestowardthedatasetwhichincludesonlythelarge energy differences(green). Table6-6.Orbitalspaceandperformanceformolecularuor ine. cc-pVXZ a [e,o]FS[e,o]AS%ReducedSpeedup b Abs.Err. cD eErr. c D[18,28][14,16]435x0.890.19T[18,60][14,32]4725x0.310.31Q \ [18,110][14,58]4762x1.950.16 Q[18,110][14,31]72314x0.052.175 \ [18,182][14,101]4524x2.500.18 5[18,182][14,57]69511x2.490.20 a The \ superscriptstandsforresultsobtainedwiththelowestcom mon cutoff; b Thecalculationusedforthespeedbenchmarkistheenergy computationatequilibriumgeometry;c WithrespecttotheCCSDTreferencevaluesinmE hunits. 93

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TheresultsinTable 6-6 arefairlyconsistentwiththeexceptionofthecc-pVQZbasi s wheretheenergyatthedissociationlimitisofffromtheCCS DTvalueby2.11mE hand, strangely,theequilibriumenergyisclosertotheCCSDTwit hasmallerextendedactive space(0.05mE h)thanwiththelargerextendedactivespace(1.95mE h).Thisbehavior leadstotheD ebeingsignicantlybetterforthelattercalculationbuton lybecauseboth pointsarein 2mE herrorcomparedtothereference.Thereisnothingoutofthe ordinaryobservedintheconvergenceortheamplitudesinth ecc-pVQZcalculation.This istheonlyoccurrencethusfarofafailuretogetwithinamE hofthereferenceD eforno apparentreason. TheXTCCSDismuchlesssensitivetothechoiceoftheextende dactivespace sinceonlyasmallportionofit(t 1{ },andt 2{ })issaved.Forthatreason,itwill alwaysusetheminimalspaceprovidedbytheASDAandwillhav ethefastestpossible executiontimes.TheTCCSDmethodisabout500xfasterthanC CSDTwithacc-pV5Z basisbutattheexpenseofaccuracy:14.08mE habsoluteerrorand6.91mE h D eerror fromtheCCSDTreference. 6.3Ethylene 6.3.1Background Theethylenetwistisasomewhatdifferenttypeofbondbreak ingsinceonlythe -bondisbrokenwhilethe -bondremainsintact.Singlereferencemethodstend toproduceacuspinthePESatthe90 anglewhilemulti-referencemethodsshow atransitionstatewithastationarypoint.Thetwistedethy lenesinglethasbecome apopularexampleforelectronicstructuremethodswhichco ncernthemselveswith describingnon-dynamiccorrelation.Somethingassimplea sintroducingasecond determinantintoaHartree-Fockcalculationwillproducer easonablePESandtorsional barrierheightprovidedthebasissetislargeenough[ 146 160 ].Smallbasisset calculationshavebeenperformedwithSpin-Flipmethods[ 77 ],VOO-CCSD[ 78 ],various multi-referencemethodsinHoffmannetal.[ 61 ],multi-referenceFockspace[ 115 ], 94

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andEquation-of-Motionformalism[ 116 ].Largerbasissetcalculationswhichshould produceexperimentalresultshavebeenperformedwithCASP T2byMolina etal. [ 109 ], withmulti-referenceperturbationtheory[ 159 ],andwithstate-specicmulti-reference perturbationtheory[ 102 ].Notallofthecalculationsaboverelaxthebond-lengths andanglesalongthePESwhichdoesnotproducehighlyaccura tevaluesforthe barrierheight[ 50 ].Infact,manyexperimentalpublicationsonethylenestat ethatthe changesbondlengthsandanglesareimportantindeterminin ganaccuratebarrier height[ 18 35 107 155 ]. Thechallengeofcomputinglarge-basisenergyvaluesforth eethylenebarrier ofrotationaretwofold.First,thenumberofbasisfunction sgrowsquicklyduetoits size.Second,thegeometryoftheCCHangleandtheCCandCHbo ndsneedtobe relaxedwhichputsfurthertimepressureinobtaininghighlevelcorrelatedresultswith largebasissets.Sincenon-iterativecorrections(suchas CCSD(T))donotproducethe correctPESbehavioratthetwistedgeometry,sometypeofin expensiveactivespace methodisdesirable.Therewereseveralmethodsproposedin therecentliteraturewhich obtaincorrectPESbehaviorwithsmallbasissets,howevert heirperformancewithlarge basissetsremainuntested. TheTCCSDmethodfacesaparticularchallengewithethylene (asitwillwithmost polyatomicmolecules).Sinceonlythe[2,2]determinantis solvedintherststep,any couplingbetweentheelectronsbelongingtothisdetermina ntandtherestofthespace areexcluded.Normally,forsinglebondbreakingofadiatom ic,itcanbeassumed thatthiscouplingisnegligibleduetotherelativelysmall overlapsoftheorbitals.In polyatomicstructureswithlesssymmetry,thereismorepre valentorbitaloverlapand strongercorrelationcouplingbetweentheelectronsinthe bondbeingbrokenandother electronsinthesystem. 95

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6.3.2Results Twistedethylenepresentsachallengeforsinglereference theoryduetothe bi-radicalnatureacquiredbythecarbon-carbonbondatthe transitionstate.Also,the densityofexcitedstatesatthatpointmakesitdifcultfor allofthemethodsemployed inthisstudytoconvergetothegroundstatesolution.Itisp ossiblethatthesefeatures ofthetransitionstatealsotranslateintothephysicalrea lmsincetheexperimental measurementsofthebarrierofrotationofethyleneinthegr oundstatearenotin agreementwitheachother(seethevaluesandreferencesinT able 6.3.2 ). Inordertoobtainthecorrectenergyforthebarrierofrotat ion,r CC,r CH,and \HCHcoordinatesareoptimizedasthedihedralanglechangesfro m0degreesto90degrees. Thegeometryoptimizationisdonewitheachmethodandbasis setwherepossibleand theresultsaresummarizedinTable 6-7 .Theexperimentalgeometryattheequilibrium is:r CC = 1.339,r CH = 1.086, \HCH = 117.6 [ 57 ]. TheaverageerrorscalculatedinTable 6-7 isdonewithrespecttotheCCSDT calculationinthatbasissetintheunitsof1000 a.u.suchthatthescaleofgeometry comparisonisonthesamescaleasmE hisforenergies.WhiletheCCSD(T)method providestheexpectedprecisiontoCCSDTatequilibrium,it clearlydeviatesatthe twistedgeometry.CCSDperformsasexpected.TheTCCSDmeth odsystematically providesgeometrieswhichareslightlyclosertotheCCSDTt hanthosefromthe XTCCSDmethodbutthedifferencebetweenthetwomethodsiss mallenoughthatit couldfallwithinstatisticalerror.TheFXTCCSDgeometrie shavethesmallesterrorswith theCCSDTgeometriesacrosstheboard. ItisprohibitivelytimeconsumingtoobtaintheCCSDTgeome trieswiththecc-pVQZ basissetintheserialACESIIprogram.Itisestimatedthate achgeometryoptimization 96

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wouldrequireapproximately75days 1 Attheequilibriumgeometry,theexpectationis thattheCCSD(T)geometryisvirtuallyasgoodastheCCSDTge ometrysothatoffers acomparison.Inthecaseofthetwistedgeometrytheactives pacemethodsdescribed hereareconsideredpredictiveofwhatthetrueCCSDTvalues wouldhavebeen. Thereappearstobealackofagreementintheliteratureasto thevalueofthe rotationalbarrierofethylene(Table 6.3.2 ).Basedontheanalysisaccomplishedthusfar, themostdenitivevaluefortherotationbarrierwouldbeus ingtheFXTCCSD/cc-pVQZ geometryandenergy:110.05mE h.Inordertodetermineifthisvalueisatthe basissetlimit,asinglepointFXTCCSDenergycalculationi sperformedusingthe FXTCCSD/cc-pVQZgeometrieswithcc-pV5Zbasisset.Equati ons 6–4 and 6–3 are usedtoestimatetheCBSvalue:110.26mE hwhichmeansthatinalllikelihoodthe orbitalspacehasbeenexhausted. Thetheoreticalvaluefortherotationbarrieris10to20mE hhigherthanthe experimentalvalueslistedinTable 6.3.2 .Thesevaluesareestimatedfromvibrational spectroscopyexperimentsandwesuggestagasphaseNMRexpe rimentbedone.The challengewiththeNMRexperimentisthatonewouldneedapro beof800 Ctoreach theneededkineticrangeandthehottestcurrentprobesgono higherthan550 C 2 Theexperimentalgeometryattheequilibriumis:r CC = 1.339,r CH = 1.086, \HCH = 117.6 [ 57 ]. 1 30hoursperCCSDTsinglepoint;6pointspergeometrystep;e stimated10 geometrystepsforconvergence 2 personalcommunicationwithDr.AlexMarchioneatDuPontCe ntralResearchand Development 97

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Table6-7.Optimizedgeometryofethylene.Distancesarein andanglesareindegrees.Unitsoferrorare1000 a.u. cc-pVDZcc-pVTZcc-pVQZ ]HCCH = 0 ]HCCH = 90 ]HCCH = 0 ]HCCH = 90 ]HCCH = 0 ]HCCH = 90 CCSDTr CC1.35091.46871.33291.4513NA.NA.r CH1.09751.10281.07881.0826NA.NA. \HCH117.00116.58117.18116.76NA.NA. CCSD(T)r CC1.35071.49551.33301.47491.33081.4667r CH1.09751.10021.07871.08061.07971.0821 \HCH117.02117.68117.20117.66117.10117.30 Avg.Err.0.1511.590.159.76NA.NA. CCSDr CC1.34541.45491.32711.43471.33291.4274r CH1.09611.10201.07711.08261.07881.0839 \HCH116.92116.10117.08116.04117.18115.84 Avg.Err.1.685.371.907.10NA.NA. TCCSDr CC1.35111.46541.33241.45031.32481.4488r CH1.09751.10151.07701.08121.07781.0822 \HCH117.00116.58117.20116.90117.00116.70 Avg.Err.0.450.810.521.24NA.NA. XTCCSDr CC1.34741.46121.32931.44481.32491.4413r CH1.09601.10161.07711.08131.07801.0828 \HCH116.94116.32117.08116.50116.98116.38 Avg.Err.1.233.051.522.89NA.NA. FXTCCSDr CC1.34931.46121.33121.45001.32891.4454r CH1.09731.10271.07931.08351.08081.0859 \HCH116.96116.55117.16116.74117.10116.56 Avg.Err.0.550.280.510.51NA.NA. 98

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Table6-8.Barrierofrotationforethylene.Energies arereportedinmE hunits. cc-pVDZcc-pVTZcc-pVQZ CCSD130.78137.53136.81 CCSD(T)107.66116.38118.42 CCSDT103.94109.24NA.TCCSD110.97116.09114.04 XTCCSD104.81109.38109.24 FXTCCSD105.00108.74110.05 experimentalvalues103 a 95.3 b 90.8 c theoreticalvalues100.7 d 91.9 e 108.2 f 104.8 f a FromRef.[ 35 ]; b FromRef.[ 155 ]; c FromRef.[ 18 ]; d FromRef.[ 160 ]; e FromRef.[ 109 ]; f FromRef.[ 159 ]. 6.4Bicyclo[1,1,0]butane 6.4.1Background Theisomerizationofbicyclo[1,1,0]butaneto trans -buta-1,3-dienehasbeen experimentallyshowntoproceedthroughaconrotarytransi tionstate[ 145 ].There isalsoaforbiddenpathwayto trans -buta-1,3-dienethroughahigh-energybiradical disrotarytransitionstate[ 70 96 118 ].BothtransitionstatesareshowninFigure 6-8 Recently,KinalandPiecuchpublishedtheactivationbarri erenergiesforboth pathwaysusingaCASSCF(10,10)/cc-pVDZgeometries[ 70 ]followedbyavarietyof multi-referencemethodsforthecomputationoftheenergyg ap[ 96 ].Thezero-point energies(ZPE)andthegeometrieswasdeterminedwithCASSC F(10,10)/cc-pVDZin allcalculationsinthelatterpaper.Theisomerizationpat hwayofbicyclo[1,1,0]butane hassincethenbeenexaminedwithtwo-electronreduceddens itymatrix(2-RDM)by Mazziotti[ 106 ]andwithoptimalmultireference-diffusionMonteCarlo(O MR-DMC)by BernerandLchow[ 15 ]. Onlytheconrotarypathwayactivationbarrierandthegeome tryofbicyclo[1,1,0]butane areisexperimentallyknown.Whileitappearstobepossible toobtainacorrect geometryandbarrierenergyoftheconrotarypathwaywithsi nglereferencemethods[ 15 99

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Figure6-8.Selectgeometryparametersofisomerizationof bicyclo[1,1,0]butaneand thetransitionstates.ThebondlengthsareinAngstromsand energiesarein kcal/mol. 100

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96 ]duetoonlyasmallmulti-referencecharacterofthecorres pondingtransitionstate, thedisrotarytransitionstateappeartorequireamuti-ref erecetreatmenttocorrectly predictitsgeometryandenergy[ 15 ]. Basedonthegeometryresultsfromtheethylenestudy,wepro posethatthe FXTCCSDoptimizedgeometrywiththelargestpossiblebasis setwillprovidean excellentgeometryandenergyforthedisrotarytransition state. 6.4.2Results Thegeometriesofbicyclo[1,1,0]butane,conrotaryTSandd isrotaryTSare optimizedusingtheFXTCCSDmethodwithcc-pVDZandcc-pVTZ basissets.The orbitalspacepartitioningisdoneinaccordancewiththeAS DAdescribedpreviously. VibrationalfrequenciesandtheZPEarecalculatedwithccpVDZbasissetbutnot higherduetocomputationaltimeconstraints 3 .Asinglepointcalculationisdonewith cc-pVQZbasissetateachcc-pVTZgeometryforabettertotal energy. Inthecc-pVDZbasissetthefullspaceof15occupiedand71vi rtualorbitalsis reducedto11occupiedand30virtualactivespaceorbitals. Inthecc-pVTZbasisset thefullspaceof15occupiedand189virtualorbitalsisredu cedto11occupiedand35 virtualactivespaceorbitals.Inthecc-pVQZbasissetthef ullspaceof15occupiedand 385virtualorbitalsisreducedto11occupiedand68virtual activespaceorbitals.These orbitalspacesaregeneratedautomaticallybytheASDA.The drasticreductioninthe virtualspacecreatesthepossibilitythataCCSDT-quality result,unobtainablewithout accesstoalargeHPCcluster,maybeacquiredinamatterofda ysonamodernlaptop computer. Thegeometrywheretheexperimentalvaluesareavailable[ 30 ]isinexcellent agreementwiththecc-pVTZvaluesandweassumethatthetran sitionstategeometries 3 600singlepointcalculationsneededforaHessianwithappr oximately2hoursper pointwouldtakeabout50days 101

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Table6-9.Selectbondlengthsofbicyclo[1,1,0]butaneiso mers.The carbon-carbonbondlengthsofbicyclo[1,1,0]butaneandco nrotary anddisrotarytransitionstatesarelistedin. FXTCCSD/FXTCCSD/CASSCF(10,10) a/Experiment b cc-pVDZcc-pVTZcc-pVDZ bicyclo[1,1,0]butaneC 4 C 31.5091.4941.5221.498C 1 C 31.5091.4941.5281.497 conrotaryTSC 4 C 31.4291.4011.455NA.C 3 C 21.5331.5221.527NA.C 1 C 31.5621.5661.563NA.C 1 C 21.4691.4381.497NA. disrotaryTSC 4 C 31.4791.4551.511NA.C 3 C 21.4821.4581.511NA.C 1 C 31.5851.5911.561NA.C 1 C 21.4771.4541.504NA. a TheCASSCF(10,10)/cc-pVDZvaluesarefrom[ 70 ]. b Theexperimentalvaluesarefrom[ 30 ]. areofasimilarquality.Selectgeometryparametersaresho wninFigure 6-8 andin Table 6.4.2 BasedontheresultspresentedinTable 6.4.2 ,theCASSCF(10,10)tendstoward overestimatingthecarbonbondlengthsforthissystembutt heFXTCCSD/cc-pVTZ geometryisingoodagreementwiththeavailableexperiment algeometry.Furthermore, thevibrationalfrequenciesandZPE(discussedindetailin thenextsection)froma cc-pVDZcalculationareinexcellentagreementwiththeexp erimentalvalues[ 157 ]. HarmonicZPEvaluesarenotparticularlysensitivetobasis setchoiceforasystemof comparablesize[ 74 ]sothecc-pVDZvalueisusedwithcondencethatthediffere nce betweentheavailableZPEvaluesandcc-pVTZvaluesisunder akcal/mol. Thecc-pVDZZPEvaluescalculatedwithanharmonically-sca ledfrequencies forbicyclo[1,1,0]butaneandconrotaryanddisrotarytran sitionstatesare:52.34 kcal/mol,49.47kcal/moland48.86kcal/molrespectively. TheexperimentalZPEfor bicyclo[1,1,0]butaneis52.43kcal/mol.Thisleadstothef ollowingactivationbarriers (H)usingcc-pVTZenergyandgeometry:42.16kcal/molforthe conrotarypathwayand 59.28kcal/molforthedisrotarypathway.Theseresultsare summarizedinTable 6-10 102

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SinglepointFXTCCSDenergycalculationsaredoneattheFXT CCSD/cc-pVTZ geometrieswithcc-pVQZbasissettoobtainthevaluesforco nrotaryanddisrotary transitionstates:41.54kcal/moland58.78kcal/molwitha nharmonically-scaled frequencies. Selectprevioustheoreticalresultsfortheconrotarypath wayinclude:E=42.2 kcal/molandH=41.5kcal/molPT2FenergyatMCSCF(10,10)geometrywith6 -31G* basis(fordetailssee[ 118 ]);H=41.3kcal/molCR-CC(2,3)/CBSwithCASSCF(10,10) geometryandZPE[ 96 ];H=41.2kcal/mol2-RDM/6-311G**[ 106 ];H=40.4kcal/mol OMR3-DMC[ 15 ].Forthedisrotarypathway:E=60.3kcal/molandH=56.3kcal/mol PT2F//MCSCF(10,10)/6-31G*basis[ 118 ];H=67.1kcal/molCR-CC(2,3)/CBSwith CASSCF(10,10)geometryandZPE[ 96 ];H=55.7kcal/mol2-RDM/6-311G**[ 106 ];H =58.7kcal/molOMR3-DMC[ 15 ]. TheexperimentalvalueforHfortheconrotarypathwayis41(2.5)[ 145 ]butthe valueforthedisrotarypathwayisunknown. Allofthepreviousmethodsfromliterature,experimentalv alueandthevaluesin thisworkareingoodagreementregardingtheactivationene rgybarrieroftheconrotary path.However,themoredifculttocomputedisrotarypathi snotasconsistent.Our valueforHisinbetteragreementwiththevalueofNguyenandGordonas opposedto LutzandPiecuch.However,onemustpointoutthatthisagree mentislikelyfortuitous duetoadifferenceinthebasissetsemployedaswellasasubs tantialdifference betweentheFXTCCSD/cc-pVTZgeometriesandtheMCSCF(10,1 0)/6-31G*geometries whichwereclosertotheCASSCF(10,10)/cc-pVDZgeometries in[ 70 ]. InordertondsomeconformationoftheFXTCCSDresult,aCCSD(T)calculation wasperformed.TheCCSD(T)methodiscapableinovercomingsomeofthe multi-referencedifcultiesoftheregularCCSD(T)method [ 151 152 ].Table 6-10 includestheresultsfromtheCCSD(T)calculationwhichisinagreementwiththe FXTCCSDresult.TheEvalueisthedifferencebetweentotalenergyandtheHvalue 103

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Table6-10.Energeticsofbicyclo[1,1,0]butaneisomeriza tion.Allenergyunitsarein kcal/mol.Thecc-pVQZvaluesareenergieswithcc-pVTZgeom etries. FXTCCSD/FXTCCSD/FXTCCSD/CCSD(T)/CCSD(T)/ cc-pVDZcc-pVTZcc-pVQZcc-pVDZcc-pVTZ bicyclo[1,1,0]butaneZPE54.2454.16 ZPEsc52.3452.30 ThroughE42.7945.0344.2843.5945.37 conrotaryTSZPE51.3951.43H39.9342.1841.5640.8642.64 ZPEsc49.4749.56Hsc39.9242.1641.5440.8542.63 ThroughE58.7762.7662.2260.8664.91 disrotaryTSZPE50.7450.83H55.2659.2658.8157.5361.59 ZPEsc48.8648.90Hsc55.2959.2858.7857.4661.51 includescorrectionforZPE.TheZPEisonlycalculatedwith thecc-pVDZbasisand thethescsubscriptdenotesananharmonically-scaledvalue.The2-R DMactivation barrierenergyforthedisrotaryTSaswellastheOMR3-DMCar ebothveryclosetothe FXTCCSDandevenCCSD(T)valueswhendonewithacomparablebasisset. 6.5FrequenciesandZPE VibrationalfrequenciescalculatedinACESIIarebasedona harmonicpotential. Themostinexpensivewaytocorrectforthedifferenceinene rgybetweenharmonic energylevelsandanharmonicenergylevelsistoempiricall yderiveascalingfactor usingexperimentalfrequencies.Thescalingfactorvaries fordifferentmethodsand basissets.Awidevarietyofthescalingfactorsisavailabl eontheComputational ChemistryComparisonandBenchmarkDataBase(CCCBDB) 4 TheZPEwhichisonehalfthesumofallfrequenciesshouldals obescaled. Sincetheactivespacemethodspresentedherearenovelandt hereisnoentryonthe CCCBDBforthem,wederiveourownscalingfactorsbasedonth eexamplesathand: ethyleneandbicyclo[1,1,0]butanewiththecaveatthatonl ythosefrequencieswhich 4 http://cccbdb.nist.gov/ 104

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Table6-11.Vibrationalfrequenciesofbicyclo[1,1,0]but aneisomerization. ExperimentFXTCCSD/cc-pVDZFXTCCSD/cc-pVDZ-scaled BICBICCONDISBICCONDIS 423408*408*657661*409*352*661*409*352*737755578*386*724555386*838861637579826611555839873704682*837675682*909924761*746*886761*746*935942*890*842942*890*808980986*918893986*880856 10631083*9459941083*90695410811104100910181059968977108111051037*1034*10601037*1034*1092113610501094113610071049111011711115114411231069109711721186120111561138115311091261129312721226124112201176126613351361138112801305132514851487147914701426141914101501152815081481146514461421293530873089309829612963297229693090315431082964302629813044319631773136306630483008304431983189319830683059306831203261319532193128306430883131327332683252314031353120 scalingfactor0.959 ZPE52.4354.2451.3950.7452.3449.4748.86 aredominatedbyastretcharetobeusedinthedetermination ofthescalingfactorand subsequentlyscaled.Bendingandtorsionalfrequenciesge nerallyhavepotentialswhich arebestdescribedbytrigonometricfunctionsandtherefor eshouldnotbeunilaterally scaled. Table 6-11 showsallofthevibrationalfrequencies,scalingfactorsa ndZPEvalues usedinthedeterminationofthescaledHvalues.Theexperimentalvaluesare from[ 157 ].The*signifymodeswhichareidentiedtobeprimarilyben dingand/or twistingandthusarenotscaled. 105

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Thescalingfactorisderivedby:scaling factor =P~exp ~calc ~2 calc(6–5) whereonlythepredominantlystretchingfrequenciesareus edinthesummation.This isrepeatedwiththeCCSD(T)forwhichthescalingfactoris0.960.Finally,thes ame procedureisrepeatedwiththeethylene(experimentalfreq uencyvaluesfrom[ 86 ])at FXTCCSD/cc-pVDZandthescalingfactorremainsconsistent at0.959. 6.6Conclusion Singlereferencecoupledclustermethodsareshowntohavet heabilityto successfullyaddressmulti-referenceproblemsasseenint hetwocasesdemonstrated inthisworkwhenCCSDTprovidesagoodtreatmentofasingleb onddissociation. ForhigherMRcharacterexamplessuchasozonevibrationalf requenciesordouble bonddissociation,onewouldexpectthesamecanbeaccompli shedwithCCSDTQand higher. Wehaveoutlinedaprocedurethatbringsthecostofhighorde rcoupledcluster calculationdownconsiderablywithoutasignicantlossof accuracy,sothatmore challengingmulti-referenceproblemscannowbetackledwi thalargeenoughbasisset toyieldusefulresults. Theactivespaceselectionprocedureisautomatedandwillp erformwithverylittle interventionfromtheuser.Webelieve,philosophically,t hatthisisthebestwaytoassure atrulypredictivetheory,notbiasedtotheusers'chemical knowledgeorexperiencewith quantumchemistrysoftware. Atthispoint,thereisreasontobelievethatthecontractio nsusedintheDunning basissetsyieldalessthanoptimalenergymanifoldofthevi rtualorbitalswhichtends tocausetheASDAtomakeearliercutoffsthannecessary.Whe ncomparedtoalarge ANObasisset,thestatisticalanalysisyieldsmoreconsist entresults:noearlycutoffs andnolargeamplitudeswhichgooutsidetheextendedspace. 106

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CHAPTER7 CONCLUSIONSANDFUTUREWORK Thereareseveralnovelapproachesdescribedinthisdisser tationwhichgreatlyaid chemistsinperformingcomputationallydemandingcalcula tionatalowercostbutwith littletonosacriceinaccuracy.STEOMisawellestablishe dmethodwhichtendsto beunderutilizedduetoitslimitedavailability.Futureim plementationofSTEOMinthe parallelACESIIIsoftwarewillgreatlyincreaseitsusabil ityandpresenceintheliterature. Theanalyticttingtovibtrationalnormalmodesfollowedb yasolutionofthe Hamiltonianinadiscretevariablerepresentationproduce sagoodvibronicspectrum, evenwhensomeofthenormalmodesaredissociative;whichis themainadvantageof theDVRapproach.Themode-modecouplingishiddeninthepot entialenergysurfaces andisintroducedperturbativelyandanalytically. TheactivespacepartitioningoftheHilbertspacemaynotbe anewconcept. However,anautomaticapproachwithoutuserbiasorknowled geofchemistryisnovel. Generally,thosewhouseCASandCCSDtqmethodstendtopicka ctivespacesbased ontheirpriorknowledgeofthesystemoruntilthetheoretic alandtheexperimental resultsreachagreement.TheASDAprovidesaphilosophical lydifferentapproachby usingthestatisticalanalysisoftheHilbertspacetodeter minetheactivespacecutoffs. TheASDAissuperblydemonstratedwithseveralsimplebutno velactivespace methodsinwhichthecorrelationintheactivespaceissolve dwithCCSDTandthe correlationinthefullspaceissolvedwithCCSDsubjecttot heconstantamplitudes fromtheCCSDTcalculation.Thisapproachyieldsexcellent CCSDT-qualityresults withoutincurringtheCCSDTcost.Thesemethodsareeasilya mendabletomanyother combinations:CCSDTQ/CCSD,CCSDT-1/CCSD,CCSD/MBPT(2), etc .,alldepending onthelevelofaccuracydesired.Furthermore,theASDAallo wstheHilbertspacetobe partitionedinmanylayersprovidingapossibilitytoperfo rmaCCSDQ/CCSD/MBPT(2) calculationforalargesystemandexpecttheresultstobecl osetoCCSDTQinquality. 107

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Finally,aninterestingapplicationofallofthetechnique sproposedinthisdissertation istousetheFXTCCSDmethodtocalculatethegroundstatePES alongwithSTEOM-CCSD tocomputetheexcitedstatePESinordertoobtaindissociat iveabsorptioncross sections.TheRHF-UHFPESdiscontinuityobservedintheNaO Hexamplewouldbe mitigatedanditshouldleadtosignicantlybetteranalyti cpotentialts.Thenalproduct wouldbeastand-aloneDVRprogramwritteninahigh-levella nguage(suchasRor MatLab)whichwouldincludeinputparsing,curve-tting,s olvingtheDVRvibrational Hamiltonian,andgeneratingtheplotsofthevibroniccross section. 108

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REFERENCES [1]Agmon,Noam.“ElementaryStepsinExcited-StateProton Transfer.” TheJournal ofPhysicalChemistryA 109(2005).1:13–35.PMID:16839085. [2]Andrews,L.“Ultraviolet-absorptionstudiesofalkali -metalatomoxygenmolecule matrixreaction.” JournalofMolecularSpectroscopy 61(1976).3:337–345. [3]Bartlett,RodneyJ.“Coupled-clusterapproachtomolec ularstructureandspectra: asteptowardpredictivequantumchemistry.” JournalofPhysicalChemistry 93 (1989).5:577. [4]Bartlett,RodneyJ. Coupledclustertheory:Anoverviewofrecentdevelopments Singapore:WorldScienticPublishingCo.Ltd.,1995. [5]Bartlett,RodneyJ.andMusia,Monika.“Coupled-clust ertheoryinquantum chemistry.” Rev.Mod.Phys. 79(2007):291–352. [6]Bartlett,RodneyJ.andNooijen,Marcel.“Anewmethodfo rexcitedstates: Similaritytransformedequation-of-motioncoupled-clus tertheory.” Journalof ChemicalPhysics 106(1997):6441–6448. [7]Bartlett,RodneyJ.andPurvis,GeorgeD.“Electroncorr elationinlargemolecules withmany-bodymethods.” AnnalsoftheNewYorkAcademyofSciences 367 (1981).1:62–82. [8]Bartlett,RodneyJ.andPurvisIII,G.D.“Many-bodypert urbationtheory, coupled-pairmany-electrontheory,andtheimportanceofq uadrupoleexcitations forthecorrelationproblem.” InternationalJournalofQuantumChemistry 14 (1978).5:561. [9]Bartlett,RodneyJ.andPurvisIII,G.D.“Molecularappl icationsofcoupledcluster andmany-bodyperturbationmethods.” PhysciaScripta 21(1980):255. [10]Bartlett,RodneyJ.andStanton,JohnF. Applicationsofpost-Hatree-Fock methods:atutorial .NewYork:VCHPublishers,1994. [11]Bartlett,RodneyJ.,Watts,J.D.,Kucharski,S.A.,and Noga,J.“Non-iterative fth-ordertripleandquadrupleexcitationenergycorrect ionsincorrelated methods.” ChemicalPhysicsLetters 165(1990).6:513–522. [12]Becke,AxelD.“Density-functionalthermochemistry. III.Theroleofexact exchange.” TheJournalofChemicalPhysics 98(1993).7:5648–5652. [13]Bene,JanetDelandJaff,H.H.“UseoftheCNDOMethodin Spectroscopy.III. MonosubstitutedBenzenesandPyridines.” TheJournalofChemicalPhysics 49 (1968).3:1221–1229. 109

PAGE 110

[14]Berger,R.,Fischer,C.,andKlessinger,M.“Calculati onoftheVibronicFine StructureinElectronicSpectraatHigherTemperatures.1. BenzeneandPyrazine.” JournalofPhysicalChemistryA 102(1998):7157–7167. [15]Berner,R.andLchow,A.“IsomerizationofBicyclo[1. 1.0]butanebyMeansofthe DiffusionQuantumMonteCarloMethod.” JournalofPhysicalChemistryA 114 (2010):13222–13227. [16]Bode,BrettM.andGordon,MarkS.“Macmolplt:agraphic aluserinterfacefor GAMESS.” JournalofMolecularGraphicsandModelling 16(1998).3:133–138. [17]Bolovinos,A.,Tsekeris,P.,Philis,J.,Pantos,E.,an dAndritsopoulos,G.“Absolute vacuumultravioletabsorptionspectraofsomegaseousazab enzenes.” Journalof MolecularSpectroscopy 103(1984):240–256. [18]Borrelli,RaffaeleandPeluso,Andrea.“Thevibration alprogressionsoftheN–>V electronictransitionofethylene:Atestcaseforthecompu tationofFranck-Condon factorsofhighlyexiblephotoexcitedmolecules.” TheJournalofChemicalPhysics 125(2006).19:194308. [19]Brucker,G.A.andKelley,D.F.“Excitedstateintermol ecularprotontransferin matrixisolatedbeta-naphthol/ammoniacomplexes.” TheJournalofChemical Physics 90(1989).10:5243–5251. [20]Buenker,RobertandPeyerimhoff,Sigrid.“Individual izedcongurationselection inCIcalculationswithsubsequentenergyextrapolation.” TheoreticalChemistry Accounts:Theory,Computation,andModeling(TheoreticaC himicaActa) 35 (1974):33–58. [21]Bytautas,Laimutis,Nagata,Takeshi,Gordon,MarkS., andRuedenberg,Klaus. “AccurateabinitiopotentialenergycurveofF[sub2].I.No nrelativisticfullvalence congurationinteractionenergiesusingthecorrelatione nergyextrapolation byintrinsicscalingmethod.” TheJournalofChemicalPhysics 127(2007).16: 164317. [22]Calvert,JackG.andPittsJr.,JamesN. Photochemistry .NewYork:JohnWiley& Sons,1966. [23]ChaseJr.,M.W. J.Phys.Chem.Ref.Data (1998).9:1. [24]Cheshnovsky,OriandLeutwyler,Samuel.“Excited-sta teprotontransferinneutral microsolventclusters:naphthol(NH3)n.” ChemicalPhysicsLetters 121(1985).1-2: 1–8. [25]Cheshnovsky,OriandLeutwyler,Samuel.“Protontrans ferinneutralgas-phase clusters:alpha-Naphthol[center-dot](NH[sub3])[subn] .” TheJournalofChemical Physics 88(1988).7:4127–4138. 110

PAGE 111

[26]Chutjian,A.,Hall,R.I.,andTrajmar,S.“Electron-im pactexcitationofH[sub2]O andD[sub2]Oatvariousscatteringanglesandimpactenergi esintheenergy-loss range4.2–12eV.” TheJournalofChemicalPhysics 63(1975).2:892–898. [27]Cizek,J.“Useoftheclusterexpansionandthetechniqu eofdiagramsin calculationsofcorrelationeffectsinatomsandmolecules .” AdvancesinChemical Physics 14(1969):35. [28]Cizek,J.,Paldus,J.,andShavitt,I.“Correlationpro blemsinatomicandmolecular systems.IV.Extendedcoupled-pairmany-electrontheorya nditsapplicationtothe boranemolecule.” PhysicalReviewA:Atomic,Molecular,andOpticalPhysics 5 (1972).3:50. [29]Colbert,DanielTandMiller,WilliamH.“Anoveldiscre tevariablerepresentation forquantumscatteringviatheS-matrixKohnmethod.” JournalofChemical Physics 96(1992):1982–1991. [30]Cox,KentW.,Harmony,MarlinD.,Nelson,Gordon,andWi berg,K.B.“Microwave SpectrumandStructureofBicyclo[1.1.0]butane.” TheJournalofChemicalPhysics 50(1969).5:1976–1980. [31]Coxon,JohnA.andHajigeorgiou,PhotosG.“TheB1+andX 1+ElectronicStates ofHydrogenFluoride:ADirectPotentialFitAnalysis.” J.Phys.Chem.A 110 (2006).19:6261. [32]Daigoku,Kota,ichiIshiuchi,Shun,Sakai,Makoto,Fuj ii,Masaaki,andHashimoto, Kenro.“Photochemistryofphenol–(NH[sub3])[subn]clust ers:Solventeffectona radicalcleavageofanOHbondinanelectronicallyexciteds tateandintracluster reactionsintheproductNH[sub4](NH[sub3])[subn-1](n<= 5).” TheJournalof ChemicalPhysics 119(2003).10:5149–5158. [33]David,O.,Dedonder-Lardeux,C.,andJouvet,C.“Isthe reanExcitedStateProton Transferinphenol(or1-naphthol)-ammoniaclusters?Hydr ogenDetachmentand TransfertoSolvent:Akeyfornon-radiativeprocessesincl usters.” International ReviewsinPhysicalChemistry 21(2002).3:499–523. [34]Dessent,CarolineE.H.andMijller-Dethlefs,Klaus. “Hydrogen-Bondingand vanderWaalsComplexesStudiedbyZEKEandREMPISpectrosco py.” Chemical Reviews 100(2000).11:3999–4022. [35]Douglas,JohnE.,Rabinovitch,B.S.,andLooney,F.S.“ KineticsoftheThermal Cis-TransIsomerizationofDideuteroethylene.” TheJournalofChemicalPhysics 23(1955).2:315–323. [36]Droz,Thierry,Knochenmuss,Richard,andLeutwyler,S amuel.“Excited-state protontransferingas-phaseclusters:2-Naphthol(NH3n). ” TheJournalof ChemicalPhysics 93(1990).7:4520–4532. 111

PAGE 112

[37]DunningJr.,ThomH.“Gaussianbasissetsforuseincorr elatedmolecular calculations.I.Theatomsboronthroughneonandhydrogen. ” JournalofChemical Physics 90(1989):1007–1023. [38]Evangelista,FrancescoA.“Alternativesingle-refer encecoupledcluster approachesformultireferenceproblems:Thesimpler,theb etter.” TheJournalofChemicalPhysics 134(2011).22:224102. [39]Evangelista,FrancescoA.,Allen,WesleyD.,Schaefer ,HenryF.,andIII.“Coupling termderivationandgeneralimplementationofstate-speci cmultireference coupledclustertheories.” TheJournalofChemicalPhysics 127(2007).2:024102. [40]Evangelista,FrancescoA.,Prochnow,Eric,Gauss,Jr gen,Schaefer,HenryF., andIII.“Perturbativetriplescorrectionsinstate-speci cmultireferencecoupled clustertheory.” TheJournalofChemicalPhysics 132(2010).7:074107. [41]Feller,David,Peterson,KirkA.,andHill,J.Grant.“O ntheeffectivenessof CCSD(T)completebasissetextrapolationsforatomization energies.” TheJournal ofChemicalPhysics 135(2011).4:044102. [42]Feller,DavidandSordo,JoseA.“PerformanceofCCSDTf ordiatomicdissociation energies.” TheJournalofChemicalPhysics 113(2000).2:485–493. [43]Flaud,J.M.,Peyret,C.C.,Johns,J.W.C.,andCarli,B. “Thefarinfrared spectrumofH[sub2]O[sub2].Firstobservationofthestagg eringofthelevels anddeterminationofthecisbarrier.” JournalofChemicalPhysics 91(1989): 1504–1510. [44]Fleisher,A.J.,Morgan,P.J.,andPratt,D.W.“Charget ransferbyelectronic excitation:Directmeasurementbyhighresolutionspectro scopyinthegasphase.” TheJournalofChemicalPhysics 131(2009).21:211101. [45]Flowers,BradleyA,Stanton,JohnF,andSimpson,Willi amR.“Wavelength DependenceofNitrateRadicalQuantumYieldfromPeroxyace tylNitrate Photolysis:ExperimentalandTheoreticalStudies.” JournalofPhysicalChemistry A 111(2007):11602–11607. [46]Foresman,JamesB.,Head-Gordon,Martin,Pople,JohnA .,andFrisch,MichaelJ. “Towardasystematicmolecularorbitaltheoryforexciteds tates.” TheJournalof PhysicalChemistry 96(1992).1:135–149. [47]Frster,T. Z.Elektrochem 54(1949):42. [48]Furche,FilippandAhlrichs,Reinhart.“Adiabatictim e-dependentdensity functionalmethodsforexcitedstateproperties.” TheJournalofChemicalPhysics 117(2002).16:7433–7447. 112

PAGE 113

[49]Ghose,KeyaB.,Piecuch,Piotr,andAdamowicz,Ludwik. “Improvedcomputational strategyforthestate-selectivecoupled-clustertheoryw ithsemi-internaltriexcited clusters:PotentialenergysurfaceoftheHFmolecule.” TheJournalofChemical Physics 103(1995).21:9331–9346. [50]Giroux,L.,Back,M.H.,andBack,R.A.“Acommentonther otationalisomerization ofethylene.” ChemicalPhysicsLetters 154(1989).6:610–612. [51]Granucci,Giovanni,Hynes,JamesT.,Millo,Philippe ,andTran-Thi,Thu-Hoa.“A TheoreticalInvestigationofExcited-StateAcidityofPhe nolandCyanophenols.” JournaloftheAmericanChemicalSociety 122(2000).49:12243–12253. [52]Gurvich,V.,Bergman,G.A.,Gorokhov,N.,andVungman, V.S.“Thermodynamic PropertiesofAlkaliMetalHydroxides.Part1.LithiumandS odiumHydroxides.” JournalofPhysicalChemistryReferenceData 25(1996):1211–1276. [53]Halkier,Asger,Helgaker,Trygve,Jrgensen,Poul,Kl opper,Wim,Koch,Henrik, Olsen,Jeppe,andWilson,AngelaK.“Basis-setconvergence incorrelated calculationsonNe,N2,andH2O.” ChemicalPhysicsLetters 286(1998):243– 252. [54]Hariharan,P.C.andPople,J.A.“Theinuenceofpolari zationfunctionson molecularorbitalhydrogenationenergies.” Theoreticachimicaacta 28(1973): 213–222. [55]Harrison,RobertJ.,Fitzgerald,GeorgeB.,Laidig,Wi lliamD.,andBarteltt, RodneyJ.“AnalyticMBPT(2)secondderivatives.” ChemicalPhysicsLetters 124 (1986).3:291–294. [56]Henseler,Debora,Tanner,Christian,Frey,Hans-Mart in,andLeutwyler,Samuel. “Intermolecularvibrationsof1-naphthol(NH3)andd3-1-n aphthol(ND3)intheS0 andS1states.” TheJournalofChemicalPhysics 115(2001).9:4055–4069. [57]Herzberg,G. InfraredandRamanSpectraofPolyatomicMolecules .NewYork: VanNostrandReinhold,NewYork,1945. [58]Hineman,M.F.,Brucker,G.A.,Kelley,D.F.,andBernst ein,E.R.“Excited-state protontransferin1-naphthol/ammoniaclusters.” TheJournalofChemicalPhysics 97(1992).5:3341–3347. [59]Hino,Osamu,Kinoshita,Tomoko,Chan,GarnetKin-Lic, andBartlett,RodneyJ. “Tailoredcoupledclustersinglesanddoublesmethodappli edtocalculationson molecularstructureandharmonicvibrationalfrequencies ofozone.” TheJournal ofChemicalPhysics 124(2006).11:114311. [60]Hirao,K.“MultireferenceMller-Plessetmethod.” ChemicalPhysicsLetters 190 (1992).3-4:374–380. 113

PAGE 114

[61]Hoffmann,MarkR.,Datta,Dipayan,Das,Sanghamitra,M ukherjee,Debashis, gnesSzabados,Rolik,Zoltn,andSurjn,PterR.“Compar ativestudyof multireferenceperturbativetheoriesforgroundandexcit edstates.” TheJournalof ChemicalPhysics 131(2009).20:204104. [62]Huber,K.J.andHerzberg,G. MolecularSpectraandMolecularStructure: ConstantsofDiatomicMolecules .NewYork:VanNostrandReinhold,NewYork, 1979. [63]Humphrey,SusanJ.andPratt,DavidW.“Acid–basechemi stryinthegasphase: Thetrans-1-naphthol[center-dot]NH[sub3]complexinits S[sub0]andS[sub1] electronicstates.” TheJournalofChemicalPhysics 104(1996).21:8332–8340. [64]Ivanov,VladimirV.andAdamowicz,Ludwik.“CASCCD:Co upled-clustermethod withdoubleexcitationsandtheCASreference.” TheJournalofChemicalPhysics 112(2000).21:9258–9268. [65]Jacoby,Christoph,Hering,Peter,Schmitt,Michael,R oth,Wolfgang,and Kleinermanns,Karl.“InvestigationsofOHN-andNHO-typeh ydrogen-bonded clustersbyUVlaserspectroscopy.” ChemicalPhysics 239(1998).1-3:23–32. [66]Jenkin,MichaelE.,Boyd,AndrewA.,andLesclaux,Robe rt.“Peroxy RadicalKineticsResultingfromtheOH-InitiatedOxidatio nof1,3-Butadiene, 2,3-Dimethyl-1,3-ButadieneandIsoprene.” JournalofAtmosphericChemistry 29 (1998):267–298. [67]Johnson,J.R.,Jordan,K.D.,Plusquellic,D.F.,andPr att,D.W.“Highresolution S1<–S0uoescenceexcitationspectraofthe1-and2-hydrox ynaphthalenes. Distinguishingthecisandtransrotamers.” TheJournalofChemicalPhysics 93 (1990).4:2258–2273. [68]Jouvet,C.,Lardeux-Dedonder,C.,Richard-Viard,M., Solgadi,D.,andTramer, A.“Reactivityofmolecularclustersinthegasphase:proto n-transferreactionin neutralphenol-(ammonia)nandphenol-(ethanamine)n.” TheJournalofPhysical Chemistry 94(1990).12:5041–5048. [69]Kimura,K.andNagakura,S.“Vacuumultra-violetabsor ptionspectraofvarious mono-substitutedbenzenes.” MolecularPhysics 9(1965).2:117–135. [70]Kinal,ArmaganandPiecuch,Piotr.“ComputationalInv estigationofthe ConrotatoryandDisrotatoryIsomerizationChannelsofBic yclo[1.1.0]butane toButa-1,3-diene:ACompletelyRenormalizedCoupled-Clu sterStudy.” The JournalofPhysicalChemistryA 111(2007).4:734–742.PMID:17249766. [71]Kinoshita,Tomoko,Hino,Osamu,andBartlett,RodneyJ .“Coupled-cluster methodtailoredbycongurationinteraction.” TheJournalofChemicalPhysics 123(2005).7:074106. 114

PAGE 115

[72]Knochenmuss,Richard.“Excited-stateprotontransfe rin1-naphthol(NH3n complexes:thethresholdsizeisn=4.” ChemicalPhysicsLetters 293(1998).3-4: 191–196. [73]Knochenmuss,RichardandFischer,Ingo.“Excited-sta teprotontransferin naphthol/solventclusters:thecurrentstateofaffairs.” InternationalJournalof MassSpectrometry 220(2002).2:343–357. [74]Kolmann,StephenJ.andJordan,MeredithJ.T.“Methoda ndbasisset dependenceofanharmonicgroundstatenuclearwavefunctio nsandzero-point energies:ApplicationtoSSSH.” TheJournalofChemicalPhysics 132(2010).5: 054105. [75]Koput,Jacek.“Anabinitiostudyontheequilibriumstr uctureandtorsional potentialenergyfunctionofhydrogenperoxide.” ChemicalPhysicsLetters 236 (1995):516–520. [76]Kou,Zhuangfei,Shen,Jun,Xu,Enhua,andLi,Shuhua.“H ybridcoupledcluster methods:Combiningactivespacecoupledclustermethodswi thcoupledcluster singles,doubles,andperturbativetriples.” TheJournalofChemicalPhysics 136 (2012).19:194105. [77]Krylov,AnnaI.andSherrill,C.David.“Perturbativec orrectionstothe equation-of-motionspin–ipself-consistenteldmodel: Applicationto bond-breakingandequilibriumpropertiesofdiradicals.” TheJournalofChemical Physics 116(2002).8:3194–3203. [78]Krylov,AnnaI.,Sherrill,C.David,Byrd,EdwardF.C., andHead-Gordon,Martin. “Size-consistentwavefunctionsfornondynamicalcorrela tionenergy:Thevalence activespaceoptimizedorbitalcoupled-clusterdoublesmo del.” TheJournalof ChemicalPhysics 109(1998).24:10669–10678. [79]Kuchitsu,K. StructureofFreePolyatomicMolecules-BasicData .Berlin:Springer Netherlands,1998. [80]Ku s,Tomasz,Lotrich,VictorF.,andBartlett,RodneyJ.“Para llelimplementation oftheequation-of-motioncoupled-clustersinglesanddou blesmethodand applicationforradicaladductsofcytosine.” TheJournalofChemicalPhysics 130 (2009).12:124122. [81]Larsen,N.W.“Microwavespectraofthesixmono-13C-su bstitutedphenolsand ofsomemonodeuteratedspeciesofphenol.Completesubstit utionstructureand absolutedipolemoment.” JournalofMolecularStructure 51(1979).0:175–190. [82]LeRoy,R.J.“LEVEL8.0,AComputerProgramforSolvingt heRadial SchrdingerEquationforBoundandQuasiboundLevels.” 115

PAGE 116

[83]Lee,EdmondPFandWright,TimothyG.“HeatsofFormatio nofNaOHand NaOH+:IonizationEnergyofNaOH.” JournalofPhysicalChemistryA 106 (2002):8903–8907. [84]Lee,YoonS.,Kucharski,StanislawA.,andBartlett,Ro dneyJ.“Acoupledcluster approachwithtripleexcitations.” TheJournalofChemicalPhysics 81(1984): 5906–5912. [85]Lee,Young-Shin,Yu,Hyunung,Kwon,Oh-Hoon,andJang, Du-Jeon. “Photo-inducedproton-transfercycleof2-naphtholinfau jasitezeolitic nanocavities.” Phys.Chem.Chem.Phys. 10(2008):153–158. [86]Lerberghe,D.Van,Wright,I.J.,andDuncan,J.L.“High -resolutioninfrared spectrumandrotationalconstantsofethylene-H4.” JournalofMolecularSpectroscopy 42(1972).2:251–273. [87]Li,XiangzhuandPaldus,Josef.“Reducedmultireferen ceCCSDmethod:An effectiveapproachtoquasidegeneratestates.” TheJournalofChemicalPhysics 107(1997).16:6257–6269. [88]Li,XiangzhuandPaldus,Josef.“Reducedmultireferen cecoupleclustermethod. II.ApplicationtopotentialenergysurfacesofHF,F[sub2] ,andH[sub2]O.” The JournalofChemicalPhysics 108(1998).2:637–648. [89]Li,XiangzhuandPaldus,Josef.“General-model-space state-universal coupled-clustermethod:excitationenergiesofwater.” MolecularPhysics 104 (2006).5-7:661–676. [90]Light,J.C.,Hamilton,I.P.,andLill,J.V.“Generaliz eddiscretevariable approximationinqunatummechanics.” JournalofChemicalPhysics 82(1985): 1400–1409. [91]Loch,MartaW.,Lodriguito,MaricrisD.,Piecuch,Piot r,andGour,JeffreyR.“Two newclassesofnon-iterativecoupled-clustermethodsderi vedfromthemethod ofmomentsofcoupled-clusterequations.” MolecularPhysics 104(2006).13-14: 2149–2172. [92]Lotrich,V.F.,Ponton,J.M.,Perera,A.S.,Deumens,E. ,Bartlett,R.J.,andSanders, B.A.“Superinstructionarchitectureofpetascaleelectro nicstructuresoftware:the story.” MolecularPhysics 108(2010).21-23:3323–3330. [93]Lotrich,VictorF.,Flocke,N.,Ponton,M.,Yau,A.D.,P erera,A.,Deumens,E.,and Bartlett,R.J.“Parallelimplementationofelectronicstr uctureenergy,gradient,and Hessiancalculations.” TheJournalofchemicalphysics 128(2008):194104. [94]Lchow,ArneandAnderson,JamesB.“AccuratequantumM onteCarlo calculationsforhydrogenuorideandtheuorineatom.” TheJournalofChemical Physics 105(1996).11:4636. 116

PAGE 117

[95]Luter,H.“Silverman,B.W.:DensityEstimationforSt atisticsandDataAnalysis. ChapmanandHall,London ASNewYork1986.” BiometricalJournal 30 (1988).7:876–877. [96]Lutz,JesseJ.andPiecuch,Piotr.“Extrapolatingpote ntialenergysurfacesby scalingelectroncorrelation:Isomerizationofbicyclobu tanetobutadiene.” The JournalofChemicalPhysics 128(2008).15:154116. [97]Lyakh,DmitryI.andBartlett,RodneyJ.“Anadaptiveco upled-clustertheory: @CCapproach.” TheJournalofChemicalPhysics 133(2010).24:244112. [98]Lyakh,DmitryI.,Ivanov,VladimirV.,andAdamowicz,L udwik.“State-specic multireferencecomplete-active-spacecoupled-clustera pproachversusother quantumchemicalmethods:dissociationoftheN2molecule. ” MolecularPhysics 105(2007).10:1335–1357. [99]Lyakh,DmitryI.,Musia,Monika,Lotrich,Victor,and Bartlett,RodneyJ. “Multireferencenatureofchemistry:Thecoupled-cluster view.” Chem.Revs. (2012).112:182–243. [100]Madronich,S.,Hastie,D.R.,Ridley,B.A.,andSchiff ,H.I.“Calculationsof theTemperatureDependenceoftheNO2PhotodissociationCo efcientinthe Atmosphere.” JournalofAtmosphericChemistry 1(1984):151–157. [101]Mahapatra,UttamSinha,Chattopadhyay,Sudip,andCh audhuri,RajatK. “Molecularapplicationsofstate-specicmultireference perturbationtheoryto HF,H2O,H2S,C2,andN2molecules.” TheJournalofChemicalPhysics 129 (2008).2:024108. [102]Mahapatra,UttamSinha,Chattopadhyay,Sudip,andCh audhuri,RajatK. “Second-orderstate-specicmultireferenceMllerPless etperturbation theory:Applicationtoenergysurfacesofdiimide,ethylen e,butadiene,and cyclobutadiene.” JournalofComputationalChemistry 32(2011).2:325–337. [103]Marston,ClayC.andBalint-Kurti,G.,Gabriel.“TheF ouriergridHamiltonian methodforboundstateeigenvaluesandeigenfunctions.” JournalofChemical Physics 91(1989):3571–3576. [104]Matsumoto,Y,Ebata,T,andMikami,N.“Structuresand vibrationsof 2-naphthol(NH3)n(n=1-3)hydrogen-bondedclustersinves tigatedbyIR/UV double-resonancespectroscopy.” JournalofMolecularStructure 552(2000):257 –271. [105]Mayor,E.andVelasco,A.M.“Photodissociationofthe d(0,0)andd(1,0)bands ofnitricoxideinthestratosphereandthemesosphere:Amol ecular-adapted quantumdefectorbitalcalculationofphotolysisratecons tants.” Journalof GeophysicalResearch 112(2007):D13304. 117

PAGE 118

[106]Mazziotti,D.A.“EnergyBarriersintheConversionof Bicyclobutaneto gauche-1,3-ButadienefromtheAnti-HermitianContracted Schro Ldinger Equation.” JournalofPhysicalChemistryA 112(2008):13684–13690. [107]Merer,A.J.andMulliken,RobertS.“Ultravioletspec traandexcitedstatesof ethyleneanditsalkylderivatives.” ChemicalReviews 69(1969).5:639–656. [108]Mikami,Naohiko,Okabe,Akihiro,andSuzuki,Itaru.“ Photodissociationofthe hydrogen-bonded[phenol-ammonia]+heterodimerion.” TheJournalofPhysical Chemistry 92(1988).7:1858–1862. [109]Molina,Vicent,Merchan,Manuela,Roos,BjornO.,and Malmqvist,Per-Ake.“On thelow-lyingsingletexcitedstatesofstyrene:atheoreti calcontribution.” Phys. Chem.Chem.Phys. 2(2000):2211–2217. [110]Monnerville,MandRobbe,JM.“Opticalpotentialcoup ledtodiscretevariable representationforcalculationsofquasiboundstates:App licationinteractiontothe COpredissociating.”101(1994):7580–7591. [111]Mota,R.,Parata,R.,Giuliani,A.,Hubin-Franskin, M.-J.,Lourenaso,J.M.C., Garcia,G.,Hoffmann,S.V.,Mason,N.J.,Ribeiro,P.A.,Rap oso,M.,and Limato-Vieira,P.“WaterVUVelectronicstatespectroscop ybysynchrotron radiation.” ChemicalPhysicsLetters 416(2005):152–159. [112]Mulliken,RobertS.“MolecularCompoundsandtheirSp ectra.III.TheInteraction ofElectronDonorsandAcceptors.” TheJournalofPhysicalChemistry 56(1952): 801–822. [113]Bartlett,RodneyJandMusia,Monika.“Coupled-clus tertheoryinquantum chemistry” ReviewofModernPhysics 79(2007):291–352. [114]Musia,MonikaandBartlett,RodneyJ.“Charge-trans ferseparabilityand size-extensivityintheequation-of-motioncoupledclust ermethod:EOM-CCx” TheJournalofChemicalPhysics 134(2011):034106. [115]Musia,MonikaandBartlett,RodneyJ.“Multi-refere nceFockspace coupled-clustermethodintheintermediateHamiltonianfo rmulationforpotential energysurfaces.” TheJournalofChemicalPhysics 135(2011):044121. [116]Musia,Monika,Perera,Ajith,andBartlett,RodneyJ .“Multireference coupled-clustertheory:Theeasyway.” TheJournalofChemicalPhysics 134 (2011).11:114108. [117]Nagakura,S.andGouterman,M.“EffectofHydrogenBon dingontheNear UltravioletAbsorptionofNaphthol.” TheJournalofChemicalPhysics 26(1957).4: 881–886. [118]Nguyen,KietA.andGordon,MarkS.“IsomerizationOfB icyclo[1.1.0]butaneto Butadiene.” JournaloftheAmericanChemicalSociety 117(1995).13:3835–3847. 118

PAGE 119

[119]Nix,MichaelG.D.,Devine,AdamL.,Cronin,Brd,Dixo n,RichardN.,andAshfold, MichaelN.R.“Highresolutionphotofragmenttranslationa lspectroscopystudies ofthenearultravioletphotolysisofphenol.” TheJournalofChemicalPhysics 125 (2006).13:133318. [120]Nooijen,Marcel.“ElectronicExcitationSpectrumof s-Tetrazine:An Extended-STEOM-CCSDStudy.” TheJournalofPhysicalChemistryA 104 (2000).19:4553–4561. [121]Nooijen,MarcelandHazra,Anirban.“VIBRON-AProgra mforVibronicCoupling andFranck-CondonCalculations.WithcontributionsfromJ .F.StantonandK. Sattelmeyer.”2003. [122]Olsen,Jeppe,Roos,BjrnO.,rgensen,PoulJ,andrge nAa.Jensen,HansJ. “Determinantbasedcongurationinteractionalgorithmsf orcompleteand restrictedcongurationinteractionspaces.” TheJournalofChemicalPhysics 89(1988).4:2185–2192. [123]Paldus,J. MethodsinComputationalMolecularPhysics .NewYork:Plenum Press,1992. [124]Piecuch,Piotr.“Active-spacecoupled-clustermeth ods.” MolecularPhysics 108 (2010).21-23:2987–3015. [125]Piecuch,Piotr,awA.Kucharski,Stanis,andBartlet t,RodneyJ.“Coupled-cluster methodswithinternalandsemi-internaltriplyandquadrup lyexcitedclusters: CCSDtandCCSDtqapproaches.” TheJournalofChemicalPhysics 110 (1999).13:6103–6122. [126]Piecuch,Piotr,Kowalski,Karol,Pimienta,IanS.O., andMcguire,MichaelJ. “Recentadvancesinelectronicstructuretheory:Methodof momentsof coupled-clusterequationsandrenormalizedcoupled-clus terapproaches.” InternationalReviewsinPhysicalChemistry 21(2002).4:527–655. [127]Pino,G.,Gregoire,G.,Dedonder-Lardeux,C.,Jouvet ,C.,Martrenchard, S.,andSolgadi,D.“Aforgottenchannelintheexcitedstate dynamicsof phenol-(ammonia)clusters:hydrogentransfer.” Phys.Chem.Chem.Phys. 2 (2000):893–900. [128]Plane,J.M.C.“Theroleofsodiumbicarbonateinthenu cleationofnoctilucent clouds.” AnnalesGeophysicae 18(2000):807–814. [129]Plusquellic,D.F.,Tan,X.-Q.,andPratt,D.W.“Acidbasechemistryinthegas phase.Thecis-andtrans-2-naphthol(NH3)complexesinthe irS0andS1states.” TheJournalofChemicalPhysics 96(1992).11:8026–8036. 119

PAGE 120

[130]Pope,FrancisD,Hansen,JaronC,Bayes,KyleD,Friedl ,RandallR,andSander, StanleyP.“Ultravioletabsorptionspectrumofchlorinepe roxide,ClOOCl.” The journalofphysicalchemistry.A 111(2007):4322–32. [131]PurvisIII,G.D.andBartlett,RodneyJ.“Afullcouple d-clustersinglesanddoubles model:theinclusionofdisconnectedtriples.” JournalofChemicalPhysics 76 (1982).4:1910–1918. [132]Puzari,Panchanan,Swathi,RottiS,Sarkar,Biplab,a ndAdhikari,Satrajit.“A quantum-classicalapproachtothephotoabsorptionspectr umofpyrazine.” The Journalofchemicalphysics 123(2005):134317. [133]Rauhut,Guntram.“Efcientcalculationofpotential energysurfacesforthe generationofvibrationalwavefunctions.” TheJournalofchemicalphysics 121 (2004):9313–22. [134]Redmon,LynnT.,Purvis,GeorgeD.,andBartlett,Rodn eyJ.“Accuratebinding energiesofdiborane,boranecarbonyl,andborazanedeterm inedbymany-body perturbationtheory.” JournaloftheAmericanChemicalSociety 101(1979).11: 2856–2862. [135]Rosta,EdinaandSurjn,PterR.“Two-bodyzerothord erHamiltoniansin multireferenceperturbationtheory:TheAPSGreferencest ate.” TheJournalof ChemicalPhysics 116(2002).3:878–890. [136]Sadlej,AndrzejJ.“Medium-sizepolarizedbasissets forhigh-level-correlated calculationsofmolecularelectricproperties.” TheoreticaChimicaActa 79(1991): 123–140. [137]Schmidt,MichaelW.,Baldridge,KimK.,Boatz,JerryA .,Elbert,StevenT., Gordon,MarkS.,Jensen,JanH.,Koseki,Shiro,Matsunaga,N ikita,Nguyen, KietA.,Su,Shujun,Windus,TheresaL.,Dupuis,Michel,and Montgomery, JohnA.“Generalatomicandmolecularelectronicstructure system.” Journalof ComputationalChemistry 14(1993).11:1347–1363. [138]Sears,JohnS.,Sherrill,C.David,andKrylov,AnnaI. “Aspin-completeversion ofthespin-ipapproachtobondbreaking:Whatistheimpact ofobtainingspin eigenfunctions?” TheJournalofChemicalPhysics 118(2003).20:9084–9094. [139]Self,DanielE.andPlane,JohnM.C.“Absolutephotoly siscross-sectionsfor NaHCO3,NaOH,NaO,NaO2andNaO3:implicationsforsodiumch emistryin theuppermesosphere.” PhysicalChemistryChemicalPhysics 4(2002):16–23. [140]Shen,Jun,Xu,Enhua,Kou,Zhuangfei,andLi,Shuhua.“ Acoupledcluster approachwithahybridtreatmentofconnectedtripleexcita tionsforbond-breaking potentialenergysurfaces.” TheJournalofChemicalPhysics 132(2010).11: 114115. 120

PAGE 121

[141]Siebrand,Willem,Zgierski,MarekZ.,Smedarchina,Z orkaK.,Vener,Mikhail,and Kaneti,Jose.“Thestructureofphenol-ammoniaclustersbe foreandafterproton transfer.Atheoreticalinvestigation.” ChemicalPhysicsLetters 266(1997).1-2:47 –52. [142]Sobolewski,AndrzejL.andDomcke,Wolfgang.“Photoi nducedElectronand ProtonTransferinPhenolandItsClusterswithWaterandAmm onia.” TheJournal ofPhysicalChemistryA 105(2001).40:9275–9283. [143]Solgadi,D.,Jouvet,C.,andTramer,A.“Resonance-en hancedmultiphoton ionizationspectraandionizationthresholdsofphenol-(a mmonia)nclusters.” The JournalofPhysicalChemistry 92(1988).12:3313–3315. [144]Solntsev,KyrilM.,Huppert,Dan,andAgmon,Noam.“So lvatochromismof beta-Naphthol.” TheJournalofPhysicalChemistryA 102(1998).47:9599–9606. [145]Srinivasan,R.,Levi,A.A.,andHaller,I.“TheTherma lDecompositionof Bicyclo[1.1.0]butane.” TheJournalofPhysicalChemistry 69(1965).5:1775–1777. [146]Staemmler,Volker.“NoteonopenshellrestrictedSCF calculationsforrotation barriersaboutC-Cdoublebonds:Ethyleneandallene.” TheoreticalChemistry Accounts:Theory,Computation,andModeling(TheoreticaC himicaActa) 45 (1977):89–94.10.1007/BF00552543. [147]Stanton,J.F.,JGauss,J.T.,Perera,S.A.,Watts,J.D .,Yau,A.D.,Nooijen,M., Oliphant,N.,Szalay,P.G.,Lauderdale,W.J.,Gwaltney,S. R.,Beck,S.,Balkova, A.,Bernholdt,D.E.,Baeck,K.K.,Rozyczko,P.,Sekino,H., Huber,C.,Pittner,J., andCencek,W.“ACESIIisAProgramProductOfTheQuantumThe oryProject UniversityOfFlorida.packagesIncludedAreVMOL(J.Almlo f,P.R.Taylor); VPROPS(P.Taylor);ABACUS(T.Helgaker,H.J.Aa.Jensen,P. Jrgensen,J. Olsen,P.R.Taylor);HONDO/GAMESS(M.W.Schmidt,K.K.Bald ridge,J.A.Boatz, S.T.Elbert,M.S.Gordon,J.J.Jensen,S.Koseki,N.Matsuna ga,K.A.Nguyen,S. Su,T.L.Windus,M.Dupuis,J.A.Montgomery.” [148]Stanton,JohnFandBartlett,RodneyJ.“Theequationo fmotioncoupled-cluster method.Asystematicbiorthogonalapproachtomolecularex citationenergies, transitionprobabilities,andexcitedstateproperties.” TheJournalofChemical Physics 98(1993).9:7029. [149]Suppan,Paul.“Invitedreviewsolvatochromicshifts :Theinuenceofthemedium ontheenergyofelectronicstates.” JournalofPhotochemistryandPhotobiology A:Chemistry 50(1990).3:293–330. [150]Tanner,Christian,Henseler,Debora,Leutwyler,Sam uel,Connell,LeslieL.,and Felker,PeterM.“Structuralstudyofthehydrogen-bonded1 -naphthol(NH32 cluster.” TheJournalofChemicalPhysics 118(2003).20:9157–9166. 121

PAGE 122

[151]Taube,AndrewG.andBartlett,RodneyJ.“Improvingup onCCSD(T):Lambda CCSD(T).I.Potentialenergysurfaces.” TheJournalofChemicalPhysics 128 (2008).4:044110. [152]Taube,AndrewG.andBartlett,RodneyJ.“Improvingup onCCSD(T):Lambda CCSD(T).II.Stationaryformulationandderivatives.” TheJournalofChemical Physics 128(2008).4:044111. [153]Tolbert,LarenM.andSolntsev,KyrilM.“Excited-Sta teProtonTransfer:From ConstrainedSystemstoSuperPhotoacidstoSuperfastProto nTransfer.” Accounts ofChemicalResearch 35(2002).1:19–27.PMID:11790085. [154]Trost,Barbara,Stutz,Jochen,andPlatt,Ulrich.“UV -absorptioncrosssections ofaseriesofmonocyclicaromaticcompounds.” AtmosphericEnvironment 31 (1997).23:3999–4008. [155]Wallace,R.“Thetorsionalenergylevelsofethylene: Are-evaluation.” Chemical PhysicsLetters 159(1989).1:35–36. [156]Watanabe,K.andZelikoff,M.“AbsorptionCoefcient sofWaterVaporinthe VacuumUltraviolet.” J.Opt.Soc.Am. 43(1953).9:753–754. [157]Wiberg,KennethB.,Waddell,ShermanT.,andRosenber g,RobertE.“Infrared intensities:bicyclo[1.1.0]butane.Anormalcoordinatea nalysisandcomparison withcyclopropaneand[1.1.1]propellane.” JournaloftheAmericanChemical Society 112(1990).6:2184–2194. [158]Widmark,PerOlof,Malmqvist,PerAke,andRoos,Bjorn O.“Densitymatrix averagedatomicnaturalorbital(ANO)basissetsforcorrel atedmolecularwave functions.” TheoreticaChimicaActa 77(1990):291–306. [159]Witek,HenrykA.,Nakano,Haruyuki,andHirao,Kimihi ko.“Multireference perturbationtheorywithoptimizedpartitioning.II.Appl icationstomolecular systems.” JournalofComputationalChemistry 24(2003).12:1390–1400. [160]Wood,M.H.“Thebarriertorotationforthegroundstat eofethylene:aDCSCF approach.” ChemicalPhysicsLetters 24(1974).2:239–242. [161]Woon,DavidE.andDunningJr.,ThomH.“Gaussianbasis setsforusein correlatedmolecularcalculations.V.Core-valencebasis setsforboronthrough neon.” JournalofChemicalPhysics 103(1995):4572–4585. [162]Yang,Jie,Hao,Yusong,Li,Juan,Zhou,Chang,andMo,Y uxiang.“Acombined zeroelectronickineticenergyspectroscopyandion-paird issociationimaging studyoftheF2+structure.” TheJournalofChemicalPhysics 122(2005).13: 134308. 122

PAGE 123

[163]Yoshino,K.,Esmond,J.R.,Parkinson,W.H.,Ito,K.,a ndMatsui,T.“Absorption crosssectionmeasurementsofwatervaporinthewavelength region120to188 nm.” ChemicalPhysics 211(1996):387–391. [164]Yu,Hua-Gen.“Acoherentdiscretevariablerepresent ationmethodfor multidimensionalsystemsinphysics.” TheJournalofchemicalphysics 122 (2005):164107. [165]Zemke,WarrenT.,c.Stwalley,William,Langhoff,Ste phenR.,Valderrama, GiuseppeL.,andBerry,MichaelJ.“Radiativetransitionpr obabilitiesforall vibrationallevelsintheX 1=Sigma +stateofHF.” TheJournalofChemicalPhysics 95(1991).11:7846. [166]Zwier,TimothyS.“Thespectroscopyofsolvationinhy drogen-bondedaromatic clusters.” AnnualReviewofPhysicalChemistry 47(1996).1:205–241. 123

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BIOGRAPHICALSKETCH AnnwasbornintheUSSRin1982intheRepublicofUkraine.She spenther childhoodinBrooklyn,NYwheresheattendedtheBrooklynTe chnicalHighSchool andgraduatedwithNYShonorsin2000.AnngraduatedwithaB. A.inchemistryfrom CarnegieMellonUniversityin2003andwentontotheUnivers ityofHawai'iatManoato obtainaM.S.inchemistryunderthetutelageofDr.JohnHead In2006AnnstartedherdoctoratedegreewithDr.RodneyBarl ettintheQuantum TheoryProjectattheUniversityofFlorida.In2013Anngrad uatedwithaPh.D.in chemistryandaM.S.inelectricalandcomputerengineering .Sheiscurrentlyemployed attheIntelCorporation. 124